GIFT  OF 
Author 


THE  INDUCTION  MOTOR 

AND 

OTHER  ALTERNATING  CURRENT  MOTORS 


(Frontispiece) 


THE  INDUCTION  MOTOR 

AND 

OTHER  ALTERNATING" 
CURRENT  MOTORS 


THEIR   THEORY  AND  PRINCIPLES 
OF  DESIGN 


1  t 

B.  A.  BEHREND 

FELLOW,  AND  PAST  SENIOR  VICE  PRESIDENT,  AMERICAN  INSTITUTE  OF  ELECTRICAL  ENGINEERS  i 
FELLOW,  AMERICAN  ACADEMY  OF  ARTS  4  SCIENCES;  FELLOW,  AMERICAN  ASSOCIATION 
FOR  THE  ADVANCEMENT  OF  SCIENCE;  MEMBER,  AMERICAN  SOCIETY  OF  CIVIL  ENGI- 
NEERS AND  AMERICAN  SOCIETY  OF  MECHANICAL  ENGINEERS,  ETC. 


SECOND  EDITION 
REVISED  AND  ENLARGED 

SECOND  IMPRESSION 


McGRAW-HILL  BOOK  COMPANY,  INC. 
NEW  YORK:  370  SEVENTH  AVENUE 

LONDON :  6  &  8  BOUVERIE  ST.,  E.  C.  4 

1921 


COPYRIGHT,  1921,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY,  INC. 


PRINTED    IN   THE   UNITED   STATES   OF    AMERICA 


THE    MAPLE    FXtKSS    Y  O  K  K   FA 


•Go 

THE  GREAT  PIONEERS 

WHO  HAVE  BEEN  MY  FRIENDS 

NIKOLA  TESLA,  GISBERT  KAPP,  ANDRE  BLONDEL,  C.  E.  L.  BROWN 

THIS  BOOK  IS  AFFECTIONATELY  INSCRIBED 


/f  -I    A  O 


" Ignorance  more  frequently  begets  confidence  than  does 

knowledge." 

CHARLES  DARWIN, 
"The  Descent  of  Man,"  p.  3. 

"It  is  particularly  interesting  to  note  how  many  theorems,  even  among 
those  not  ordinarily  attacked  without  the  help  of  the  Differential  Calculus, 
have  here  been  found  to  yield  easily  to  geometrical  methods  of  the  most 
elementary  character. 

"Simplification  of  modes  of  proof  is  not  merely  an  indication  of  advance 
in  our  knowledge  of  a  subject,  but  is  also  the  surest  guarantee  of  readiness 
for  farther  progress." 

LORD  KELVIN  AND  PETER  GUTHRIE  TAIT, 
"Elements  of  Natural  Philosophy,"  p.  v. 

"The  simplicity  with  which  complicated  mechanical  interactions  may 
be  thus  traced  out  geometrically  to  their  results  appears  truly  remarkable." 

SIR  GEORGE  HOWARD  DARWIN, 

"On  Tidal  Friction,"  in  "Treatise  on  Natural  Philosophy." 
By  KELVIN  AND  TAIT,  p.  509. 

" the  absence  of  analytical  difficulties  allows  attention  to  be 

more  easily  concentrated  on  the  physical  aspects  of  the  question,  and  thus 
gives  the  student  a  more  vivid  idea  and  a  more  manageable  grasp  of  the 
subject  than  he  would  be  likely  to  attain  if  he  merely  regarded  electrical 
phenomena  through  a  cloud  of  analytical  symbols." 

SIR  JOSEPH  JOHN  THOMSON, 

"Elements  of  the  Mathematical  Theory 

of  Electricity  and  Magnetism,"  p.  vi. 

"It  is  remarkable  that  such  elementary  cases  of  Newton's  dynamics 
should  require  abstruse  considerations  for  their  explanation.  But  it  is  far 
worse  in  the  more  modern  dynamics,  with  ignoration  of  coordinates,  and 
modified  Lagrangean  functions.  Dynamics  as  visible  to  the  naked  eye 
seems  to  disappear  altogether  sometimes,  leaving  nothing  but  complicated 
algebra." 

OLIVER  HEAVISIDE, 
"Electromagnetic  Theory,"  vol.  iii,  p.  401. 

"Let  them  make  the  effort  to  express  these  ideas  in  appropriate  words 
without  the  aid  of  symbols,  and  if  they  succeed  they  will  not  only  lay  us 
laymen  under  a  lasting  obligation,  but,  we  venture  to  say,  they  will  find 
themselves  very  much  enlightened  during  the  process,  and  will  even  be 
doubtful  whether  the  ideas  as  expressed  in  symbols  had  ever  quite  found 
their  way  out  of  the  equations  into  their  minds." 

"The  Scientific  Papers"  of  JAMES  CLERK  MAXWELL, 

vol.  ii,  p.  328. 
vii 


"The  scientific  career  of  Rankine  was  marked  by  the  gradual  develop- 
ment of  a  singular  power  of  bringing  the  most  difficult  investigations  within 
the  range  of  elementary  methods." 

"The  Scientific  Papers"  of  JAMES  CLERK  MAXWELL, 
vol.  ii,  p.  663. 

"Lagrange  came  to  grief  over  the  small  conical  oscillations  of  the  spherical 
pendulum,  yet  he  could  have  saved  himself  and  detected  his  error  but  for 
the  self-imposed  restraint  of  excluding  the  diagram  from  his  "  Mecanique 
analytique."  So  it  is  curious  to  find  the  same  fashion  coming  again  in  the 
modern  school  of  pure  analytical  treatment,  of  doing  away  with  an  appeal 
to  the  visual  sense  of  a  geometrical  figure." 

SIE  GEORGE  GREENHILL, 
Nature,  April  17,  1919. 


viii 


PREFACE  TO  SECOND  EDITION 

As  indicated  in  the  preface  to  the  first  edition  of  this  little 
book,  it  owed  its  origin  to  a  series  of  lectures  delivered  at  the 
University  of  Wisconsin  in  January,  1900.  These  lectures  were 
published  in  the  Electrical  World  and  appeared  in  1901  in  book 
form  entitled  "The  Induction  Motor."  The  book  was  translated 
into  several  languages  among  others  into  French  and  German. 
The  American  edition  was  soon  exhausted  and  repeatedly 
attempts  were  made  by  myself  and  by  assistants  and  associates 
of  mine  to  revise  it,  but  though  several  agreements  were  entered 
into  between  the  publishers  and  Mr.  A.  B.  Field  and  myself  for 
a  second  edition,  other  and  more  urgent  demands  upon  our  time 
prevented  the  completion  of  the  work. 

Once  more  then,  twenty  years  after  its  first  appearance,  this 
little  book  addresses  itself  to  the  engineering  public.  The  first 
edition  contained  at  the  time  almost  entirely  new  matter  and 
almost  all  of  this  originated  with  the  author.  Tested  though  it 
was  by  the  most  careful  laboratory  work,  yet  a  certain  diffidence 
prevented  the  author  from  pressing  his  claims  to  recognition. 
Rarely  perhaps  has  any  early  work  become  so  absorbed  into  the 
texture  of  thought  of  engineers  as  the  substance  of  this  little 
book.  The  kindly  words  of  my  friend  Dr.  Addams  S.  McAllister 
in  a  presentation  copy  of  his  own  excellent  treatise  on  "  A  Iternating 
Current  Motors,"  in  which  he  says  that  to  the  present  author 
"all  writers  on  induction  motors,  and  all  students  of  induction 
motor  phenomena,  are  indebted  for  the  first  presentation  of  the 
conception  of  the  phenomena  now  considered  modern,"  I  would 
not  here  repeat — though  I  treasure  them  very  highly — were  it  not 
for  the  fact  that  as  a  perusal  of  the  introduction  may  indicate — 
my  work  is  constantly  quoted  as  done  by  others  and  these  quota- 
tions are — as  a  dispassionate  analysis  indicates — not  in  accord- 
ance with  the  plain  facts. 

The  circle  diagram  has  become  indispensable  to  the  engineer. 
Its  first  demonstration  and  proof  were  developed  by  me  in  1895. 
In  its  present  form  it  is  used  exactly  as  given  by  me  in  "The  Induc- 
tion Motor,"  New  York,  1901.  The  idea  of  the  leakage  coefficient 

ix 


X  PREFACE  TO  SECOND  EDITION 

and  its  characteristics  have  been  found  correct  and  have  been 
universally  adopted.  The  conception  of  the  single-phase  motor 
with  the  primary  exciting  belt  resolved  into  two  component 
motors  simulated  by  two  poly-phase  motors  in  series  with  opposite 
torque,  which  conception  I  worked  out  quantitatively,  has 
recently  been  commended  as  the  best  method  for  students  in  a 
paper  by  Mr.  B.  G.  Lamme*  (A.  I.  E.  E.,  April  1918).  Yet,  here 
as  elsewhere,  "A  prophet  is  not  without  honor,  save  in  his  own 
country,  and  in  his  own  house."  I  think  the  explanation  for 
this  must  be  found  in  the  tendency  of  mankind  to  prefer  to  give 
recognition  to  those  remote  from  us  rather  than  to  associates  or 
acquaintances.  It  is  easier,  for  instance,  to  name  some  one 
whom  the  students  do  not  know  and  will  not  come  in  contact 
with,  as  the  originator  of  a  certain  theory,  than  a  man  whom 
they  are  likely  to  meet  in  their  professional  relations.  Mankind, 
and  especially  professional  mankind,  is  chary  of  praise  of  its 
fellow-workers. 

An  interesting  example  of  this  is  furnished  in  the  theory  of 
the  regulation  of  alternators.  Numerous  references  in  American 
textbooks  are  made  to  "Po tier's  Method"  for  determining  the 
regulation  of  alternators.  Now,  I  believe  those  who  call  a 
certain  method  by  this  name  can  never  have  referred  to  A. 
Po  tier's  paper  "Sur  la  Reaction  d'Induit  des  Alternateurs," 
p.  133.  L'  Eclairage  Electrique,  28th  July,  1900.  Prof.  A. 
Potier  was  a  great  savant  and  a  gentleman.  His  paper  abounds 
in  carefully  selected  references.  He  claimed  no  new  method. 
He  stated  that  Mr.  Kapp  many  years  before  (about  1893  and 
since)  used  a  method  for  the  determination  of  the  regulation  of 
alternators  in  which  he  resolves  the  total  effect  of  the  armature 
currents  into  an  "armature  reaction"  component  and  a  " self- 
induction"  component,  forming  a  right  angle  triangle  in  the 
regulation  curve  in  the  case  of  zero  power  factor  He  then  refers 
to  copious  data  published  by  me  in  the  E.  T.  Z.,  in  L' Eclairage 
Electrique,  in  the  Electrical  World,  and  other  data  which  I  sent 
him  at  his  request  privately,  pointing  out  the  corroboration  of 
Kapp's  method  and  indicating — and  this  is  the  only  new  point 
in  the  paper — that  this  method  therefore  implies  identity  of  the 
zero  power  factor  regulation  curves  as  given  by  Dr.  Behn- 
Eschenburg  and  myself  for  years,  with  the  saturation  curve,  a? 

*  See  also  F.  W.  Alexanderson,  Transactions  A.  I.  E.  E.,  1918,  Part  I, 
p.  691,  692,  693. 


PREFACE  TO  SECOND  EDITION  xi 

the  two  have  been  proved  by  my  tests  to  be  equidistant  and 
displaced  from  each  other.  After  this  interesting  theoretical 
remark,  he  completes  his  paper  by  citing  Kapp  and  giving  the 
method  of  determining  the  regulation,  upon  my  experimental 
data  of  which  he  bases  his  theoretical  conclusion.  Therefore, 
the  method  of  two  components  is  to  be  designated  by  no  single 
name ;  while  the  intrinsic  importance  of  zero  power  factor  regula- 
tion was  urged  continually*  since  1896  by  myself  until  its  final 
adoption  by  the  American  Institute  in  1914! 

As  I  once  wrote  to  Mr.  Oliver  Heaviside,  quoting  Huxley, 
"  Magna  est  veritas  et  praevalebit!  Truth  is  great,  certainly,  but, 
considering  her  greatness,  it  is  curious  what  a  long  time  she  is 
apt  to  take  about  prevailing."  And  to  one  with  scholarly  inclina- 
tions, eking  out  a  livelihood  by  the  practice  of  engineering,  it  is  a 
matter  of  inward  gratification  to  see  one's  work  generally  adopted 
"  among  the  rubble  of  the  foundations  of  later  knowledge  and 
forgotten."  Remembering  that  I  was  twenty  years  old  when  I 
published  the  much  referred  to  circle  diagram  I  will  say  this  to  a 
young  reader  of  the  present  generation  by  way  of  advice:  Let 
him  not  trouble  his  head  with  recognition.  "If  truth  does  not 
prevail  in  his  time,  he  will  be  all  the  better  and  the  wiser  for 
having  tried  to  help  her.  And  let  him  recollect  that  such  great 
reward  is  full  payment  for  all  his  labor  and  pains."  (Huxley.) 

In  the  treatment  of  the  theory,  the  diagram  of  fluxes  as  devel- 
oped in  1895  by  A.  Blondel  and  used  ever  since  by  me,  has  been 
continued  throughout  this  book.  Its  simplicity  and  brilliant 
elegance  is  much  to  be  preferred  to  the  older  method  of  Kapp's, 
so  extensively  adopted  by  Steinmetz  and  others.  It  is  also 
greatly  to  be  preferred  to  the  " equivalent  circuit"  methods  which 
are  interesting  as  an  exercise  but  somewhat  artificial  and  removed 
from  intimate  contact  with  the  physical  phenomena.  On 
account  of  their  identity  with  Hopkinson's,  I  have  adopted 
Blondel's  stray  coefficients,  greater  than  one,  which  are  the  recip- 
rocals of  my  former  ones,  smaller  than  one,  in  order  to  establish 
uniformity  of  notation. 

It  is  opportune  to  say  a  few  words  on  the  subject  of  the  absence 
of  complex  algebra  in  this  little  volume.  I  have  given  this 
question  a  great  deal  of  thought.  At  first  I  intended  to  give  in 
parallel  chapters  the  results  of  the  theory  in  complex  algebraic 

*  "The  Experimental  Basis  for  the  Theory  of  the  Regulation  of  Alterna- 
tors."    By  B.  A.  Behrend,  Am.  Inst.  El.  Engrs.,  May  19,  1903. 


xii  PREFACE  TO  SECOND  EDITION 

form.  But  I  became  discouraged  in  working  out  a  number  of 
problems.  The  algebraization,  to  borrow  a  term  from  Heaviside, 
is  certainly  cumbersome,  and  one  may  be  happy  indeed  if  one 
succeeds  in  avoiding  algebraic  or  arithmetic  errors.  Page  after 
page  is  covered  with  algebraic  symbols  at  which  the  careful  and 
conscientious  calculator  looks  with  much  anxiety.  It  is  indeed 
a  beautiful  method,  this  method  of  resolution  of  directed  quantities 
into  rectangular  coordinates,  but  I  doubt  whether  it  is  suitable 
for  all  types  of  engineering  minds.  Perhaps  here  as  elsewhere,  it 
is  charitable  to  let  men  work  out  the  methods  best  suited  to 
themselves  and  not  to  press  intolerance  to  the  point  of  imposing 
one  method  upon  all.  This  is  particularly  advisable  as  a  graph- 
ical method  can,  and  should,  be  checked  by  an  algebraic  process 
and  then  the  graphical  process  is  explanatory  of,  and  elucidating, 
the  physical  process.  I  have  therefore  decided  to  omit  the  use 
of  the  symbolic  method,  and  the  reader  should  turn  to  other 
works  if  he  desires  algebraic  treatment.  It  may  be  necessary  to 
emphasize  that  the  treatment  of  the  phenomena  loses  nothing 
in  accuracy  or  elegance  by  the  adoption  of  graphic  methods  which 
have  been  used  and  advocated  by  Maxwell,  Kelvin,  Sir  George 
Darwin,  Sir.  J.  J.  Thomson,  and  others. 

The  squirrel  cage  motor  with  two  secondaries  with  different 
resistances  and  leakages  is  here  treated  graphically,  and  so  is  the 
theory  of  concatenation.  A  chapter  on  speed  regulation  of 
induction  motors  is  also  added. 

On  the  subject  of  leakage  in  induction  motors  a  great  deal  has 
been  published,  but  I  have  found  it  inadvisable  to  embody  much 
of  it  in  this  edition.  Practical  formulae  and  calculations  based 
on  them  should  be  used  sparingly,  excepting  in  the  workshop, 
and  they  should  invariably  be  based  on  personal  experience. 
One  should  not  encourage  begetting  the  formula  habit. 

The  theory  of  the  single-phase  induction  motor  has  been  given 
in  two  ways.  First,  as  originally  given  by  me  in  1897  with  the 
assumption  of  two  rotating  fields,  and  the  equivalence  of  two 
rotating  field  motors  in  series;  and  secondly,  as  first  given  by 
Potier  and  Goerges  and  beautifully  completed  by  Prof.  Sumec 
with  the  use  of  the  cross-magnetization  as  used  and  advocated 
in  this  country  by  A.  S.  McAllister. 

In  the  chapter  on  the  poly-phase  series  motor,  I  have  followed 
to  some  extent  the  brilliant  work  of  Andre  Blondel  to  whom  we 
all  are  greatly  indebted  in  every  branch  of  electrical  engineering. 


PREFACE  TO  SECOND  EDITION  xiii 

The  Heyland  compensated  motor  has  logically  received  its 
treatment  in  this  chapter,  as  it  is  a  poly-phase  shunt  motor  as 
pointed  out  and  proved  by  Blondel. 

It  seemed  unnecessary  to  treat  of  the  windings  for  induction 
motors  in  view  of  A.  M.  Dudley's  treatise  on  "Connecting 
Induction  Motors,"  McGraw-Hill  Book  Co.,  1921,  and  equally 
unnecessary  to  retain  the  two  chapters  on  design  contained  in  the 
first  edition  as  the  books  of  Mr.  H.  M.  Hobart  and  of  my  former 
assistant,  Prof.  Alexander  Miller  Gray,  *  have  supplied  this  need 
better  than  I  could  have  done. 

A  few  chapters  deal  with  such  subjects  as  the  improvement  of 
power  factor,  as  suggested  by  Leblanc  and  Kapp,  the  magnetic 
pull,  and  other  allied  subjects.  Brevity  in  the  text  has  been 
preferred  to  prolixity  as  the  lesser  of  two  evils. 

This  book  is  not  meant  to  be  a  work  of  an  encyclopedic  charac- 
ter. Nothing  that  I  could  write  could,  in  that  respect,  touch  the 
work  of  Arnold  and  LaCour.  Nor  is  it  to  take  the  place  of  such 
admirable  work  as  Alexander  Russell's  which  should  be  read  by 
every  electrical  engineer.  It  is  essentially  the  work  of  an  engi- 
neer, who  has  had  the  good  fortune  to  have  been  actively  asso- 
ciated with  the  art  of  electrical  engineering  through  almost  three 
decades  and  who  has  had  a  part  in  the  development  of  the 
machines  about  which  he  writes.  He  thus  addresses  himself 
to  his  fellow-engineers,  revealing  the  methods  which  he  has 
followed  in  the  design  and  construction  of  alternating  current 
motors,  of  which  literally  millions  of  horse-power  were  executed 
under  his  direction, 

The  design  of  electrical  machinery,  as  of  all  machinery,  is 
based  upon  intelligent  comparison  of  empirical  data,  and  the 
art  of  designing,  therefore,  cannot  be  taught  without  such  data. 
The  methods  and  principles  taught  in  this  book  aim  solely  at 
creating  means  of  effecting  such  comparisons.  To  "calculate" 
a  machine,  as  the  term  is  frequently  employed,  is  not  feasible 
and  only  principles  and  fundamentals  can  be  taught  in  school. 

No  apology  is  made  for  the  personal  references  which  occur  in 
this  book.  The  tendency  to  write  books  without  references  is 
due  largely  to  the  desire  to  avoid  the  looking-up  of  other  writers' 
papers.  The  reader  is  not  benefited  by  such  treatment,  as  he 
may  frequently  prefer  the  original  to  the  treatment  of  the  author 

*  "Induction  Motor  Design  Constants."  Electrical  World,  Dec.  30,  1911. 
"Electrical  Machine  Design,"  McGraw-Hill  Book  Co.,  1913. 


xiv  PREFACE  TO  SECOND  EDITION 

whose  book  he  is  reading.  Besides,  a  knowledge  of  the  literature 
of  our  profession  is  essential  to  an  understanding  of  the  art  and 
to  an  honest  interpretation  of  the  part  played  therein  by  our 
fellow-workers. 

My  thanks  are  due  to  my  secretary,  Miss  Gladys  Naramore, 
A.  B.,  Boston  University,  1916,  for  much  painstaking  work, 
and  to  my  friend  Dr.  Addams  Stratton  McAllister  for  his  untiring 
aid,  enthusiasm,  and  criticism.  His  friendship  has  been  an  in- 
spiration and  his  labors  in  helping  me  to  put  the  book  through 
press  are  beyond  the  rendering  of  thanks.  To  the  publishers 
thanks  are  due  for  the  successful  form  of  the  book  and  to  Mr. 
John  Erhardt  for  his  efficiency  and  for  his  patience  with  the 
author. 

An  entire  chapter  has  had  to  be  added  to  the  book  on  account 
of  a  solution  of  certain  problems  of  inversion  solved  in  a  very 
elegant  manner  by  Dr.  A.  S.  McAllister  and  communicated  to  me 
before  publication  by  him.  Thus  has  been  solved  a  problem 
with  which  I  have  coped  in  vain  these  twenty-five  years. 

To  Professor  Miles  Walker,  of  the  University  of  Manchester, 
England,  I  am  indebted  for  numerous  suggestions. 

This  little  book  now  goes  forth  as  a  sort  of  engineering  testa- 
ment of  the  author's  work  in  connection  with  the  motors  invented 
thirty-three  years  ago  by  his  friend  Mr.  Nikola  Tesla.  Great 
things  have  been  done  and  illumined  by  these  theories  and 
gigantic  engineering  feats  have  been  achieved. 

and  tho' 

We  are  not  now  that  strength  which  in  old  days 

Moved  earth  and  heaven;  that  which  we  are,  we  are; 

One  equal  temper  of  heroic  hearts, 

Made  weak  by  time  and  fate,  but  strong  in  will 

To  strive,  to  seek,  to  find,  and  not  to  yield.     (Tennyson) 

BOSTON,  MASSACHUSETTS,  B.  A.  BEHREND. 

February,  1921. 


PREFACE  TO  FIRST  EDITION 

The  literature  of  electrical  engineering  has  become  so  vast  and 
extensive  that  it  is  impossible  for  any  man  to  keep  pace  with  all 
that  is  written  on  electrical  subjects.  He  who  produces  a  new 
book  that  adds  to  the  swelling  tide  of  new  publications,  may 
justly  be  asked  for  his  credentials.  My  justification  for  writing 
this  tract  will  be  found  in  the  fact  that,  though  almost  all 
branches  of  applied  electricity  have  enlisted  the  industry  of 
authors,  the  induction  motor  has  received  comparatively  little 
attention  from  competent  engineers.  The  few  whose  experience 
and  knowledge  would  entitle  them  to  speak  with  authority  on 
this  subject  are  deterred  from  publishing  by  commercial  reasons. 

I  have  made  the  induction  motor  the  subject  of  early  and 
special  studies,  and  a  comparison  of  my  treatment  of  its  theory 
with  the  purely  analytical  theories  will  show  how  far  I  have  suc- 
ceeded in  simplifying  and  elucidating  so  complex  a  subject. 
The  graphical  treatment  of  abstruse  natural  phenomena  is 
constantly  gaining  ground,  and  I  quote  with  satisfaction  the 
words  of  so  great  a  mathematician  as  Prof.  George  Howard 
Darwin,  Fellow  of  Trinity  .College,  Cambridge,  who  says  on 
p.  509  of  the  second  volume  of  Lord  Kelvin  and  Prof.  Tait's 
Treatise  on  Natural  Philosophy  that  "the  simplicity  with  which 
complicated  mechanical  interactions  may  be  thus  traced  out 
geometrically  to  their  results  appears  truly  remarkable." 

All  through  this  little  book  I  have  endeavored  to  let  inductive 
method  check  at  every  step  the  mathematical  or  graphical 
deduction  of  the  results.  A  wide  experience  with  mono-  and 
poly-phase  alternating-current  induction  motors,  gained  at  the 
Oerlikon  Engineering  Works,  Switzerland,  has  enabled  me  to  do 
so.  Thus  the  careful  reader  who  is  willing  to  profit  by  the  experi- 
ence of  others,  will  find  many  valuable  hints  and  results  which 
he  can  turn  to  account  in  his  practice.  Many  induction  motors 
have  been  designed  on  the  principles  laid  down  in  this  little 
treatise,  and  in  no  case  has  the  theory  failed  to  answer  the 
questions  suggested  by  observation. 

The  writing  of  this  book  has  been  mainly  a  labor  of  love. 
Those  who  know  of  the  troubles,  cares  and  labor  involved  in 

xv 


xvi  PREFACE  TO  FIRST  EDITION 

writing  a  book  and  bringing  it  through  the  press,  not  to  mention 
the  sacrifice  of  personal  experience  by  publication,  will  doubtless 
be  able  to  appreciate  this  thoroughly. 

I  wish  to  thank  the  editors  of  the  Electrical  World  and  Engineer 
for  the  pains  they  have  taken  with  the  publication  of  this  book, 
and  I  must  specially  thank  Mr.  W.  D.  Weaver  for  the  encourage- 
ment he  has  always  given  to  me.  To  Mr.  T.  R.  Taltavall, 
Associate  Editor  of  Electrical  World  and  Engineer,  who  has  taken 
endless  pains  with  the  proofs  of  this  book,  I  feel  very  much 
indebted. 

The  substance  of  this  volume  was  delivered  in  January,  1900 
in  the  form  of  lectures  at  the  University  of  Wisconsin,  Madison, 
Wis.,  and  I  wish  to  thank  Prof.  John  Butler  Johnson,  Dean  of 
the  College  of  Mechanics  and  Engineering,  for  the  invitation  as 
non-resident  lecturer  which  he  extended  to  me.  To  him  and 
to  Prof.  D.  C.  Jackson  I  am  greatly  indebted  for  the  hospitality 
conferred  upon  the  stranger  within  their  gates. 

SOUTH  NORWOOD,  OHIO,  B.  A.  BEHREND. 

January,  1901. 


CONTENTS 


PAGE 

PREFACE  TO  SECOND  EDITION ix 

PREFACE  TO  FIRST  EDITION.  .    xv 


CHAPTER  I 

INTRODUCTION.     HISTORICAL 

INTRODUCTION  AND  BRIEF  SKETCH  OF  THE  HISTORY  OF  THE  THEORY  OF 

THE  INDUCTION  MOTOR 1 

THE  DEVELOPMENT  OF  THE  THEORY  OF  THE  SINGLE-PHASE  INDUCTION 

MOTOR  .  18 


CHAPTER  II 

THE  THEORY  OF  FLUXES  AND  STRAY  FIELDS 

FORMULA  FOR  INDUCED  E.  M.  F 21 

BEHREND'S  AND  BLONDEL'S  STRAY-COEFFICIENTS 23 

ELECTRIC  CIRCUITS  SIMULATING  THE  LEAKAGE  PATHS  OF  THE  MAG- 
NETIC CIRCUIT  OF  THE  INDUCTION  MOTOR 24 

THE  POLAR  DIAGRAM  FOR  CONSTANT  CURRENT 25 

THE  POLAR  DIAGRAM  FOR  CONSTANT  VOLTAGE  .  27 


CHAPTER  III 

THE  GENERAL  ALTERNATING-CURRENT  TRANSFORMER 

A.  THE  TRANSFORMER  WITH  NON-INDUCTIVE  LOAD 29 

The  Author's  Method  of  Accounting  for  Primary  Resistance .    ...  31 

Another  Method  of  Accounting  for  Primary  Resistance 33 

Accounting  for  Primary  Resistance  by  the  Method  of  Reciprocal 

Vectors 38 

The  Losses  and  Their  Representation  by  Straight  Lines 42 

The  Iron  Losses  Due  to  Hysteresis  and  Eddy  Currents 44 

B.  THE  TRANSFORMER  WITH  INDUCTIVE  LOAD 46 

C.  THE  TRANSFORMER  WITH  CAPACITY  LOAD ,  47 

xvii 


xviii  CONTENTS 

CHAPTER  IV 
THE  MCALLISTER  TRANSFORMATIONS 

PAGE 

A.  RESISTANCE  IN  SERIES  WITH  THE  MOTOR  without  CORE  Loss.    .  50 

B.  REACTANCE  IN  SERIES  WITH  THE  MOTOR  without  CORE  Loss  ....  51 

C.  IMPEDANCE  IN  SERIES  WITH  THE  MOTOR  without  CORE  Loss  ....  52 

D.  RESISTANCE  IN  SERIES  WITH  THE  MOTOR  with  CORE  Loss 54 

E.  REACTANCE  IN  SERIES  WITH  THE  MOTOR  with  CORE  Loss  ...  55 

F.  IMPEDANCE  IN  SERIES  WITH  THE  MOTOR  with  CORE  Loss  ....  55 

CHAPTER  V 
THE  ROTATING  FIELD  AND  THE  INDUCTION  MOTOR 

A.  THE  AMPERE  TURNS  AND  THE  FIELD  BELT 57 

B.  THE  E.  M.  Fs.  INDUCED  IN  THE  WINDINGS 58 

C.  THE  ELEMENTARY  THEORY  OF  THE  INDUCTION  MOTOR.    ....  64 

D.  THE  SQUIRREL  CAGE 69 

E.  THE  TORQUE  AND  SLIP  AND  THE  EQUIVALENCE  OF  MOTOR  AND 

TRANSFORMER 

The  Torque 74 

The  Slip 75 

Torque  Curves 76 

F.  HIGHER  HARMONICS  IN  THE  FIELD  BELT  AND  THEIR  EFFECT  UPON 

THE  TORQUE 77 

G.  EXPERIMENTAL  DATA 81 

H.   COLLECTION  OF  DATA 81 

CHAPTER  VI 
THE  INDUCTION  GENERATOR 

A.  THE  THEORY  OF  TORQUE  AND  SLIP 82 

B.  STABILITY 83 

C.  EXPERIMENTAL  DATA  .                  85 


CHAPTER  VII 
THE  SHORT-CIRCUIT  CURRENT  AND  THE  LEAKAGE  FACTOR 

A.  THE  SLOTS 87 

B.  THE  NUMBER  OF  SLOTS  PER  POLE 89 

C.  CHARACTERISTICS  OF  ROTOR  WINDINGS 89 

D.  TEST  DATA 91 

E.  THE  LEAKAGE  FACTOR .  94 


CONTENTS  xix 

PAGE 

F.  THE  INFLUENCE  OF  THE  AIR-GAP  UPON  THE  LEAKAGE  FACTOR  .  95 

G.  THE  INFLUENCE  OF  THE  POLE-PITCH  UPON  THE  LEAKAGE  FACTOR  .  99 

H.  THE  DIFFERENT  LEAKAGE  PATHS 101 

I.  FURTHER  EXPERIMENTAL  DATA 104 

K.  WINDING  THE  SAME  MOTOR  FOR  DIFFERENT  SPEEDS 105 

L.  DRAWBACKS  OF  A  HIGH  FREQUENCY 107 

M.  HISTORICAL  AND  CRITICAL  DISCUSSION  OF  THE  LEAKAGE  FACTOR.    .  110 

N.  BIBLIOGRAPHY  .  .114 


CHAPTER  VIII 

THE  DOUBLE  SQUIRREL-CAGE  INDUCTION  MOTOR 

ARRANGEMENT  OF  SLOTS  AND  THE  LEAKAGE  PATHS 116 

EQUIVALENT  CIRCUITS  AND  FLUX  DIAGRAM 117 

AN  EXAMPLE 119 

THE  POLAR  DIAGRAM 120 

THE  TORQUE  DIAGRAM 121 

CHAPTER  IX 

POLY-PHASE  COMMUTATOR  MOTORS 
Properties  of  Commutators 

A.  THE  ACTION  OF  THE  COMMUTATOR 124 

B.  PROPERTIES  OF  PHASE  LAG  OR  LEAD  OF  THE  POLY-PHASE  COMMU- 

TATOR   127 

C.  COMPARISON  BETWEEN  INDUCTION  MOTORS  WITH  ROTORS  SHORT- 

CIRCUITED  THROUGH  RINGS  OR  OF  THE  SQUIRREL-CAGE  TYPEJ 

AND  ROTORS  SHORT-CIRCUITED  THROUGH  SYMMETRICAL  POLY- 
PHASE BRUSHES 128 

D.  THE  REFLECTION  INTO  THE  PRIMARY  CIRCUIT  OF  THE  M.  M.  F. 

OF  THE  SECONDARY  WITH  SLIP-RING  AND  COMMUTATOR  ROTORS.   130 

E.  VARIABLE  AND  CONSTANT  SECONDARY  REACTANCE  OF  THE  COM- 

MUTATOR MOTOR 131 

F.  THE  SLIP-RING  COMMUTATOR  TYPE  AS  FREQUENCY  CHANGER  .    .   136 

CHAPTER  X 
THE  SERIES  POLY-PHASE  COMMUTATOR  MOTOR 

A.  THE  THEORY  FOR  CONSTANT  CURRENT  AND  CONSTANT  POTENTIAL 

IN  THE  IDEAL  MOTOR 137 

The  Torque 141 

The  Slip 142 

B.  THE  THEORY  FOR  CONSTANT  CURRENT  AND  CONSTANT  POTENTIAL 

IN  THE  REAL  MOTOR 143 

C.  THE  NECESSITY  OF  SATURATION  FOR  STABILITY 145 


XX  CONTENTS 

CHAPTER  XI 
THE  SHUNT  POLY-PHASE  A.  C.  COMMUTATOR  MOTOR 

PAGE 

A.  HISTORICAL  INTRODUCTION 146 

B.  THE  THEORY  OF  THE  SHUNT  POLY-PHASE  A.   C.   COMMUTATOR 

MOTOR  FOR  CONSTANT   POTENTIAL 147 

C.  DETERMINATION  OF  THE  TOTAL  PRIMARY  CURRENT 149 

D.  SPEED  REGULATION  AND  THE  SLIP 151 

E.  BIBLIOGRAPHY 152 

CHAPTER  XII 
METHODS  OF  SPEED  CONTROL 

A.  CONCATENATION 153 

B.  THE  POLY-PHASE  MOTOR  WITH  SINGLE-PHASE  SECONDARY.    .    .    .   173 

CHAPTER  XIII 
METHODS  OF  SPEED  CONTROL  (Continued) 

C.  CONCATENATION  OF  AN  INDUCTION  MOTOR  WITH  THE  COMMUTATOR 

TYPES  FOR  THE  INDUCTION  OF  A  SLIP  FREQUENCY  E.  M.  F.    .    .  176 

D.  CHANGE  OF  SPEED  BY  CHANGING  THE  NUMBER  OF  POLES.    .    .    .   179 

CHAPTER  XIV 

TYPES  OF  VARIABLE  SPEED  POLY-PHASE  COMMUTATOR 

MOTORS 

A.  THE  PLAIN  SHUNT  MOTOR 182 

B.  THE  MOTOR  OF  J.  L.  LA  COUR 183 

C.  THE  MOTOR  OF  M.  OSNOS 183 

D.  THE  MOTOR  OF  H.  K.  SCHRAGE 184 

E.  PLAIN  SHUNT  MOTOR  WITH  REGULATING  WINDING  ADDED  .    .    .  186 


CHAPTER  XV 

METHODS   OF  RAISING  THE  POWER  FACTOR   OF  INDUCTION 

MOTORS 

A.  THE  METHOD  OF  LEBLANC  USING  COMMUTATOR  MACHINES  FOR 

SECONDARY  EXCITATION 187 

B.  THE  USE  OF  A  POLYPHASE  COMMUTATOR  FOR  THE  GENERATION  OF 

LEADING  CURRENTS ••    •    •   187 


CONTENTS  xxi 

PAGE 

C.  THE  METHOD  OF  LEBLANC  INDUCING  LEADING  CURRENTS  THROUGH 

RAPID  OSCILLATION  OF  AN  ARMATURE  IN  A  MAGNETIC  FIELD  .    .   189 

D.  THE  SAME  METHOD  AS  ELABORATED  BY  G.  KAPP 195 

Bibliography 196 

E.  THE  DANIELSON-BURKE  METHOD  OF  CHANGING  THE  INDUCTION 

MOTOR  INTO  A  SYNCHRONOUS  MOTOR  .  .   197 


CHAPTER  XVI 
THE  MAGNETIC  PULL  WITH  DISPLACED  ROTOR 

A.  THE  FORMULA  OF  B.  A.  BEHREND 198 

B.  THE  ACCURATE  SOLUTION  BY  J.  K.  SUMEC 201 

BIBLIOGRAPHY 203 

CHAPTER  XVII 

THE  SINGLE-PHASE  INDUCTION  MOTOR 

A.  THE  TWO-MOTOR  THEORY. 

(a)  The  Magnetizing  and  No-load  Currents 205 

(6)  The  Currents  in  the  Armature 211 

(c)  The  Torque  and  Slip 211 

(d)  Experimental  Data 213 

(e)  Calculation  of  the   Magnetizing  Current  of  the  Single-phase 

Motor 214 

CHAPTER  XVIII 

THE  SINGLE-PHASE  INDUCTION  MOTOR  (Continued) 

B.  THE  CROSS  FLUX  THEORY 

(a)  A  General  Consideration  of  the  Theory 216 

(6)  The  Derivation  of  the  Circle  Diagram  and  the  Locus  of  the 
Primary  Currents 221 

(c)  Sumec's  Circles  for  Synchronism,  no  Load,  and  Standstill  .   224 

(d)  The  Influence  of  the  Rotor  Resistance  upon  the  Primary 
Current  Locus 224 

(e)  Equivalent  Circuits 226 

(/)  Theoretical  Considerations 229 

(00  The  Torque 229 

(h)  Mechanical  Output 231 

(i)  Rotor  Copper  Losses 231 


xxii  CONTENTS 

CHAPTER  XIX 

THE  SINGLE-PHASE  REPULSION  MOTOR 

PAGE 

A.  THE  NON-COMPENSATED  REPULSION  MOTOR 

(a)  The  General  Theory 233 

(6)  The  Speed  in  the  Diagram 237 

(c)  The  Torque 237 

(d)  The  Effect  of  the  Rotor  Resistance  upon  the  Diagram   .    .    .  238 

(e)  The  Effect  of  the  Brush  Shift 238 

(/)  Commutation 240 

B.  THE  COMPENSATED  REPULSION  MOTOR  OF  WIGHTMAN,  LATOUR,  AND 

WlNTER-ElCHBERG 

(a)  Connections 240 

(6)  The  Torque 244 

(c)  Performance 245 

(d)  Leakage 245 


CHAPTER  XX 
SINGLE-PHASE  COMMUTATOR  MOTORS 

A  Condensed  Review 

A.  VARIETY  OP  TYPES  OF  SERIES  A.  C.  COMMUTATOR  MOTORS   .    .   247 

B.  OPERATING  CHARACTERISTICS  OF  DIFFERENT  TYPES 250 

C.  METHODS  OF  IMPROVING  COMMUTATION 

(a)  Resistance  Leads  and  Limits  of  Voltage  for  Commutation     .  253 

(6)  Interpole  Connections 254 

D.  THE  SHUNT  EXCITED  A.  C.  COMMUTATOR  MOTOR 255 

E.  THE  SUPPLY  OF  SINGLE-PHASE  POWER  FROM  THREE-PHASE  SYSTEMS  258 

APPENDIX  .  .261 


LIST  OF  PORTRAITS 

PAGE 

NIKOLA  TESLA Frontispiece 

CHARLES  EUGENE  LANCELOT  BROWN 3 

ANDR£  BLONDEL 23 

GISBERT  KAPP 65 

HANS  BEHN-ESCHENBURG 113 

E.  F.  W.  ALEXANDERSON 179 

MAURICE  LEBLANC 187 

BENJAMIN  GARVER  LAMME .  247 


XXlll 


THE  INDUCTION 


CHAPTER  I 

INTRODUCTION  AND  BRIEF  SKETCH  OF  THE  HISTORY 
OF  THE  THEORY  OF  THE  INDUCTION  MOTOR 

The  Induction  Motor,  or  Rotary  Field  Motor,  was  invented  by 
Mr.  Nikola  Tesla,  in  1888,  a  year  so  "memorable  for  the  experi- 
mental corroboration  by  Hertz  of  Maxwell's  electromagnetic 
waves,  a  piece  of  work  so  shrewdly  designated  by  Oliver  Heaviside 
as  "a  great  hit."1  The  Induction  Motor  was  also  "a  great  hit," 
though  many  people  could  not  see  it. 

Engineers  almost  immediately  seized  upon  its  principles. 
Work  was  proceeding  at  Pittsburgh  under  Mr.  George  Westing- 
house,  Mr.  Tesla,  Mr.  Shallenberger,  Mr.  Scott,  and  Mr.  Lamme. 
The  first  successful  motor,  however,  embodying  in  its  design  and 
construction  those  characteristic  features  which  have  marked 
the  motor  during  its  career  of  30  years,  was  designed  most 
probably  by  Mr.  C.  E.  L.  Brown  at  the  Oerlikon  Works  in 
Switzerland  in  the  year  1890.  I  said,  "most  probably"  as  it  is 
not  impossible  that  that  brilliant  engineer,  whose  untimely 
death  we  deplore,  Mr.  Michael  von  Dolivo-Dobrowolsky,  whose 
company  at  that  time  cooperated  with  the  Oerlikon  Company, 
was  as  much  responsible  for  its  creation  as  Mr.  C.  E.  L.  Brown. 
Surely,  both  engineers  deserve  the  utmost  credit.  A  20-hp. 
motor,  built  at  the  Oerlikon  Works  and  designed  by  Mr.  C.  E.  L. 
Brown,  is  shown  in  Figs.  1  and  2.  This  motor  was  exhibited 
in  1891  at  the  Electrical  Exposition  in  Frankfort-on-the-Main. 

A  study  of  its  features  discloses  the  distributed  stator  winding, 
the  small  air-gap,  and  the  squirrel-cage  rotor,  whose  invention, 
I  believe,  is  usually  correctly  credited  to  Mr.  Dolivo-Dobrowol- 

1  "HERTZ  became  quite  Maxwellian  after  his  great  hit,  save  that,  as  I 
think,  he  attached  rather  too  much  importance,  to  the  mere  equations,  as 
the  representation  of  Maxwell's  theory,  to  the  comparative  exclusion  of 
the  experimentative  and  philosophical  basis."  OLIVER  HEAVISIDE,  "Elec- 
tromagnetic Theory,"  Vol.  iii,  p.  504,  "The  Electrician"  Printing  &  Publish- 
ing Co.  Ltd.,  London. 

1 


2 


INDUCTION  MOTOR 


sky.  This  motor  was  exhibited  in  connection  with  the  first 
alternating-current  high-voltage  power  transmission  plant  in  the 
world,  the  three-phase  30,000- volt  experimental  plant  from 


FIG.  1.— Facsimile  of  Figs.  3  and  4,  E.  T.  Z.,  Dec.  4,  1891,  of  C.  E.  L.  Brown's 
20-h.p.  three-phase  alternating  current  motor. 

Lauffen  to  Frankfort,  a  distance  of  120  km.     For  further  his- 
torical references  and  data,  I  refer  to  my  papers  in  the  Elec- 


-r*ff&&**& 


(Facing  page  2) 


INTRODUCTION  3 

trical  World  and  Engineer,  from  Nov.  16,  1901  to  March  1,  1902, 
entitled  "The  Debt  of  Electrical  Engineering  to  C.  E.  L.  Brown." 
Figure  1  is  taken  from  these  papers. 


FIG.  2. — Facsimile  of  Figs.  1  and  2,  E.  T.  Z.,  Dec.  4,  1891,  of  C.  E.  L.  Brown's 
20-h.p.  three-phase  alternating  current  motor. 

It  is  very  interesting  to  observe  that  the  industrial  develop- 
ment of  machinery,  whose  operation  is  based  upon  the  correct 
interpretation  of  scientific  theory,  rarely  proceeds  rapidly  and 
securely  until  a  method  of  interpretation  of  such  theory  has  been 


4  INDUCTION  MOTOR 

devised  which  enables  the  engineer  to  visualize  the  physical 
processes  beyond  the  complex  texture  of  a  stream  of  mathe- 
matical symbols.1 

However  valuable  the  algebraization — to  borrow  a  happy  term 
from  Heaviside — of  physical  phenomena  may  be,  it  does  not 
supply  ideas  nor  does  it  supply  usually  that  symbolic  skeleton 
into  which  the  scientific  imagination  can  weave  the  texture. 
Alternating-current  phenomena  are  very  complicated  and,  if 
quantitatively  written  out  in  equations,  they  appear  indeed  to 
be  well-nigh  incomprehensible.  A  clear  comprehension  of  alter- 
nating-current theory  was  begun  by  a  series  of  brilliant  papers 
published  in  The  Electrician,  London,  1885,  by  Thomas  H. 
Blakesley,  entitled  "Alternating  Currents  of  Electricity."  This 
series  of  10  classical  papers  discussed  for  the  first  time  alternating- 
current  phenomena  by  means  of  polar  diagrams,  often  perhaps 
erroneously  called  vector  diagrams,  as  electromotive  force  and 
current  are  in  these  cases  not  vectors  at  all  in  the  physical  sense  of 
the  term.  They  are  directive  quantities  only,  because  the  maxi- 
mum value  of  the  harmonic  wave  is  used  in  their  construction. 

I  think  the  next  landmark  in  the  development  of  the  theory 
was  made  by  Mr.  Gisbert  Kapp,  in  two  papers  originally  con- 
tributed to  the  British  Institution  of  Civil  Engineers  and  the 
Institution  of  Electrical  Engineers,  the  latter  being  printed 

1  See  also  "The  Story  of  the  Induction  Motor."  By  B.  G.  LAMME,  Jour- 
nal of  the  A.  I.E.  E.,  March,  1921. 

"The  development  of  the  Induction  Motor  being,  in  reality,  an  analytical 
problem,  it  did  not  make  much  headway  in  the  'cut  and  try'  days  of  1888 
and  1889,  when  the  Westinghouse  Company  was  undertaking  to  put  it  into 
commercial  form. 

"This  brings  the  Induction  Motor  up  to  the  present.  Its  history  has 
been  a  most  interesting  one  to  those  who  are  at  all  familiar  with  it.  To  a 
certain  extent  this  type  of  apparatus  stands  apart  in  that  its  development 
has  been  due,  almost  entirely,  to  the  analytical  engineer.  It  is  almost  impos- 
sible to  conceive  that  the  Induction  Motor  could  have  been  developed  to  its 
present  high  stage  by  ordinary  'cut  and  try'  methods.  Some  good  motors 
might  have  been  obtained  in  that  way,  but  they  would  have  been  accidents 
of  design,  instead  of  the  positive  results  of  analysis,  as  the  art  now  stands. 

"New  applications  are  continually  leading  to  new  developments  which 
are  worked  out  by  the  analytical  designer  with  an  assurance  of  success  not 
exceeded  in  any  other  branch  of  the  electrical  art.  And  the  result  of  all  the 
elaborate  theory  and  complicated  analysis  and  calculation  is  a  practical 
machine  of  almost  unbelievable  simplicity  and  reliability — a  standing 
refutation  of  the  too  common  idea  that  complexity  in  theory  leads  to 
complexity  in  results." 


INTRODUCTION  5 

in  The  Electrician,  Dec.  19,  26,  1890,  London.  This  classical 
paper  of  Mr.  Kapp's  explained  in  a  simple  and  graphical  manner 
the  interesting  phenomenon  observed  by  Sebastian  Ziani  de 
Ferranti  on  his  10,000-volt  concentric  cables  from  Deptford  to 
London.  It  is  true  that  neither  the  work  of  Blakesley  nor  that 
of  Kapp  contained  new  theories  or  new  contributions  to  the 
science,  but  in  a  sense  these  papers  accomplished  more.  The 
cumbersome  mathematical  processes  with  which  these  phenom- 
ena had  been  invested  by  mathematicians  and  physicists  of  the 
time  made  their  utilization  impracticable  if  not  impossible. 
Their  interpretation  by  means  of  the  beautiful  graphical  methods 


FIG.  3.— Fig.  12,  p.  447,  The  Electrical  World,  "Theory  of  the  Transformer," 
by  F.  Bedell  and  A.  C.  Crehore.  Primary  resistance,  no  leakage,  constant 
primary  voltage. 

of  Blakesley  and  Kapp  gave  an  impulse  to  the  entire  field  of 
electrical  engineering  which  it  could  not  have  received  without 
the  labors  of  these  men. 

In  1892,  F.  Bedell  and  A.  C.  Crehore  published  a  book  entitled 
"Alternating  Currents"  (Electrical  World  Publishers)  which  was 
followed  in  1893  by  a  series  of  articles  in  the  Electrical  World. 
In  these  works,  the  polar  diagrams  were  used  with  extreme  skill 
and  lucidity,  and  they  were  applied  to  all  manner  of  problems 
including  the  theory  of  the  alternating-current  transformer. 
In  these  papers  the  theory  of  the  constant-current  transformer 


6  INDUCTION  MOTOR 

was  developed,  showing  the  locus  of  the  primary  e.m.f.  to  be  the 
periphery  of  a  circle,  and  as  Fig.  3  I  here  reproduce  their  Fig.  12, 
p.  447,  The  Electrical  World,  June  17,  1893,  showing  for  con- 
stant primary  e.m.f.,  the  polar  diagram,  with  variation  of  the 
primary  current.  This  diagram  takes  account  of  primary  resis- 
tance but  it  does  not  take  account  of  what  is  now  usually  termed 
the  leakage. 

In  1894  came  out  the  4th  edition  of  Gisbert  Kapp's  "Electric 
Transmission  of  Energy,"  in  which  a  most  brilliant  elementary 
account  is  given  of  the  phenomena  in  induction  motors.  The 
polar  diagram  was  developed,  including  the  primary  resistance 
and  the  leakage.  This  diagram  was  given,  however,  only  for 
each  individual  point  of  the  load,  showing  no  general  solution  of 
the  variation  of  the  different  characteristic  quantities  with 
variation  of  load.  It  was  based  also  on  the  method  of  represent- 
ing leakage  through  internal  self -inductive  e.m.fs.,  which  is 
rather  cumbersome. 

In  1895,  Andre  Blondel  published  in  Eclair  age  Electrique, 
Aug.  10,  17,  24,  1895,  his  fundamental  papers  entitled  "Quel- 
ques  proprietes  ge*ne"rales  des  champs  magnetiques  tournants." 
In  these  papers  he  developed  the  theory  of  the  composition  of 
magnetic  fluxes,  including  the  leakage  fluxes,  a  method  of 
conception  which  has  since  proved  of  tremendous  value. 

In  the  same  year,  the  present  author,  utilizing  the  conception 
of  magnetic  fluxes  as  developed  by  Blondel,  proved  in  a  simple 
and  direct  manner  that  with  variation  in  load  through  change 
in  the  non-inductive  resistance  of  the  secondary  load  of  a  trans- 
former, or  through  change  in  load  on  the  shaft  of  an  induction 
motor,  with  constant  primary  e.m.f.,  the  locus  of  the  primary 
current  is  a  circle  in  the  polar  diagram,  provided  the  primary 
resultant  magnetic  field  is  constant,  which  is  the  case  if  the  primary 
resistance  of  the  transformer  can  be  neglected.  After  delivering 
a  lecture  on  the  subject  early  in  1895,  he  published  the  theory 
later  as  a  paper  on  Jan.  30,  1896,  in  Elektrotechnische  Zeitschrift, 
Berlin.  After  the  lecture,  one  of  the  learned  professors  present 
expressed  his  doubt  as  to  my  theory  being  an  exact  expres- 
sion of  the  facts,  as  he  said  the  theory  was  too  simple  to 
express  the  complicated  facts.  I,  therefore,  wished  to  test  the 
results  and  I  soon  had  an  opportunity  to  do  so  on  a  60-hp. 
Oerlikon  motor.  Testing  in  those  days  was  not  a  very  simple 
matter  and  running  a  brake  test  and  watching  Siemens  dyna- 


INTRODUCTION  7 

mometers  with  zero  reading  and  Cardew  voltmeters  was  not  as 
simple  a  procedure  as  perhaps  the  present-day  generation  may 
imagine,  spoiled  as  they  are  by  all  manner  of  ingenious  appliances 
for  simple,  direct  measurements.  When  I  felt  reasonably  sure 
that  the  theory  was  very  likely  correct,  though  corroborated  by 
only  one  test,  I  embodied  the  record  of  the  test  in  the  paper  and 
sent  it  to  the  E.  T.  Z.,  which  was  at  that  time  the  central  organ 
for  discussing  such  topics  and  there  it  lay  until  Nov.  11.  1895, 
when  I  heard  from  Mr.  Gisbert  Kapp,  who  was  then  editor 
of  the  paper,  that  he  had  accepted  it.  As  stated  before,  it  was 
printed  Jan.  30,  1896. 

While  it  lay  in  the  editorial  offices,  a  letter  came  out  in  the 
E.  T.  Z.,  p.  649,  1895,  by  A.  Heyland,  discussing  a  motor  designed 
by  Mr.  Danielson  and  applying  to  it  a  circle  locus  diagram.  In 
his  letter,  Mr.  Heyland  referred  to  his  paper  in  the  E.  T.  Z., 
Oct.  11,  1894. 

I  immediately  looked  up  Mr.  Heyland's  paper,  expecting  to 
find  therein  the  same  method  of  reasoning  and  proof  which  I 
considered  novel  in  my  paper.  Instead,  I  found  a  rather  formid- 
able array  of  lines  which  I  was  quite  unable  to  comprehend  and 
which  I  reproduce  herewith  in  facsimile,  Fig.  4.  When  I  received 
the  proofs  of  my  paper,  I  inserted  a  reference  to  Mr.  Heyland's 
letter,  E.  T.  Z.,  p.  649,  1895.  Immediately  upon  the  publication 
of  my  paper,  it  was  taken  up  by  Prof.  Andre*  Blondel  in  V  Indus- 
trie Electrique,  Feb.  25,  1896,  in  a  paper  which  begins  as  follows: 

"Le  diagramme  fondamental  des  flux  d'un  moteur  asynchrone  que 
j'ai  donne  a  diverses  reprises,  a  e"te  utilise^  recemment  d'une  maniere  fort 
heureuse  par  M.  Behrend,  grace  a  la  remarque  qu'il  a  faite  que  si 
Ton  suppose  le  F  constant  et  fixe,  Fextre'mite  du  vecteur  <£  de"crit  un 
cercle.  Get  auteur  n'a  pas  cependant  donne  encore  la  solution  complete. 
C'est  celle-ci  que  je  me  propose  d'exposer  ici  en  combinant  mes  propres 
remarques  avec  les  siennes.  La  theorie  qui  r^sulte  de  cette  collaboration 
a  distance  permet  d'embrasser  d'un  coup  d'oeil  toutes  les  conditions 
de  construction  et  de  fonctionnement,  et  constitue  a  cet  e"gard  le  meilleur 
d'une  e"tude  de"tail!6e  que  j'ai  publie*e  re"cemment." 


Mr.  Heyland  also  addressed  a  letter  to  the  E.  T.  Z.,  p.  139, 
Feb.  27,  1896,  which  begins:  "Mr.  Behrend  gives  a  very  interest- 
ing derivation  of  my  diagram  .  .  .  "  and  in  this  letter  he  claims 
his  priority.  In  a  communication  to  the  E.  T.  Z.}  p.  116,  Feb.  13, 
1896,  Prof.  A.  Blondel  writes: 


8 


INDUCTION  MOTOR 


"I  have  read  with  the  greatest  interest  the  paper  by  Mr.  Behrend 
(E.  T.  Z.,  1896,  No.  5,  p.  63)  in  which  he  applies  the  diagram  of  magne- 
tic fluxes  developed  by  me  two  years  ago  in  a  very  happy  manner  to 
the  asynchronous  motor  .  .  .  " 

In  1895,  F.  Bedell  and  A.  C.  Crehore  published  a  most  inter- 
esting and  important  paper  on  ''Resonance  in  Transformer 


FIG.  4. — Facsimile  of  Figs.  1,  2,  and  3  of  A.  Heyland's  article,  "A  Graphical 
Method  for  the  Predetermination  of  the  Transformer  and  Polyphase  Motors," 
Oct.  11,  1894,  E.  T.Z. 

Circuits,"  in  the  Physical  Review,  May-June,  1895,  Vol.  ii, 
p.  442,  in  which  the  circle  locus  of  the  primary  current  of  the 
transformer  was  clearly  and  fully  treated  by  means  of  polar 
diagrams,  including  external  inductance  in  the  secondary.  This 
paper  contains  the  complete  theory,  but  it  does  not  make  use 
of  the  identity  of  the  case  treated  with  that  of  a  transformer 
with  leakage.  In  1896  came  out  "The  Principles  of  the  Trans- 


INTRODUCTION  9 

former/'  by  Frederick  Bedell,  in  which  Fig.  123,  p.  226,  shows  the 
circular  primary  current  locus  of  a  constant-potential  transformer, 
including  leakage  and  primary  resistance,  obtained  by  the  method 
of  reciprocal  vectors  from  the  constant  current  transformer 
diagram,  which  is  easier  to  derive  than  the  constant  potential 
diagram.1 

Professor  Blondel,  in  a  letter  dated  at  Paris,  Sept.  19,  1903, 
and  published  in  No.  40,  E.  T.  Z.,  1903,  writes: 

"I  have  shown  in  the  paper  referred  to2  that  Mr.  Behrend  and 
myself  have  a  just  claim  to  many  parts  of  the  circle  diagram  of  the 
ordinary  motor.  In  reference  to  Mr.  Heyland's  article,  No.  41,  E.  T.  Z., 
1894,  so  repeatedly  brought  forward,  I  may  say  that  I  have  re-read  it 
again,  but  unfortunately  I  found  it  impossible  to  discover  a  connection 
between  his  circles  and  the  diagram  under  discussion." 

1  In  his  admirable  "Direct  and  Alternating  Current  Manual,"  2d  Ed. 
New  York,  D.  Van  Nostrand  Co.,  1916,  Dr.  BEDELL  says  on  p.  288:     ''In 
any  circuit  or  apparatus  with  constant  reactance  and  variable  power  con- 
sumption the  current  will  have  a  circle  locus  if  the  supply  voltage  is  constant 
.    .    .   This  was  first  shown  by  BEDELL  and  CREHORE  in  1892.     That  the 
induction  motor  nearly  fulfills  these  conditions  and  that  its  current  locus  is 
practically  the  arc  of  a  circle,  was  first  shown  by  HEYLAND  in  1894."     A 
footnote  states,  "E.  T.  Z.,  Oct.  11,  1894;  published  later  in  book  form  and 
translated  into  English  by  ROWE  and  HELLMUND." 

The  book  referred  to  is  a  little  volume  entitled  "A  Graphical  Treatment 
of  the  Induction  Motor"  by  ALEXANDER  HEYLAND;  translated  by  G.  H. 
ROWE  and  R.  E.  HELLMUND,  New  York,  McGraw  Publishing  Co.,  1906. 
In  this  book  HEYLAND  uses  an  entirely  different  method  from  that  given 
by  him  in  1894,  using  only  a  primary  leakage  coefficient  and  thus  obtaining 
a  simple  diagram  in  contrast  to  the  one  using  coefficients  of  mutual  and  self- 
induction  in  his  paper  of  1894.  It  must  also  be  stated  that  HEYLAND  by  no 
means  first  pointed  out  the  identity  of  the  theory  of  the  alternating  current 
transformer  and  the  induction  motor  but  this  was  first  done  by  DR.  BEHN- 
ESCHENBURG  and  by  GISBERT  KAPP  in  1893  and  1894. 

2  A.  BLONDEL,  L'Eclairaye  Electrique,  p.  137,  April  25,  1903. 

"On  me  permettra  de  rappeler,  a  ce  propos,  que  j'ai  donne"  il  y  a  plu- 
sieurs  annees  la  premiere  epure  graphique  rigoureuse  des  flux,  courants  et 
forces  electromotrices  des  moteurs  asynchrones  en  fonction  des  coefficients 
de  fuite  de  Hopkinson  et  des  coefficients  K  et  k.  (Eclairage  Electrique, 
24  aout  1894,  p.  364,  et  19  octobre,  1895,  p.  100  et  254.)  Cette  epure, 
tenant  compte  de  la  resistance  du  stator,  conduisait  £  une  courbe  repre"senta- 
tive  elliptique. 

"La  propriete  indiquee  sans  demonstration  par  HEYLAND  dans  une  lettre 
&  1'Elektrotechnische  Zeitschrift,  de  1895,  qu'en  ne"gligeant  la  resistance  du 
stator,  le  lieu  bipolaire  de  I'extr6mit6  du  triangle  de  Ii  et  I2  est  un  cercle,  a 
6te  dSmontre'e,  au  moyen  du  diagramme  des  flux,  par  BERNARD  BEHREND 


10  INDUCTION  MOTOR 

Other  references  are  interesting  as  of  historic  importance. 
Henri  Boy  de  la  Tour,  in  his  book  "The  Induction  Motor," 
translation  by  C.  0.  Mailloux,  p.  123,  writes: 

"This  method,  which  is  certainly  one  of  the  most  beautiful  applica- 
tions of  graphical  methods  to  the  solution  of  electrical  problems,  is  due 
to  the  work  of  Messrs.  A.  Blondel,  B.  A.  Behrend,  and  A.  Heyland. 

"Although  M.  Blondel  may  not  have  observed  that  a  certain  point  of 
the  diagram  should  move  on  a  circumference,  he  has,  nevertheless,  in 
our  opinion,  contributed  much  to  the  discovery  which  was  made  inde- 
pendently and  almost  at  the  same  time  by  Messrs.  B.  A.  Behrend  and 
A.  Heyland,  by  his  having  given,  ahead  of  all  other  authors,  an  exact 
analytical  study  of  the  operation  of  three-phase  motors,  based  on  a 
very  simple  diagram,  which  constitutes  the  starting-point  of  these  two 
engineers." 

G.  Kapp  says,  p.  459  of  "Dynamos,  Motors,  Alternators,  and 
Rotary  Converters,"  3d  edition,  1902: 

"For  practical  purposes  the  so-called  circle  diagram,  as  elaborated  by 
Heyland,  is  preferable.  In  the  text  I  have  chiefly  followed  Behrend' s 
work." 

And  in  the  4th  edition  he  adds  by  way  of  consolation: 

"In  the  previous  chapter  the  circle  diagram  is  obtained  substantially 
as  shown  in  the  classical  papers  of  Heyland,  Behrend,  et  al." 

Another  reference  is  found  in  the  following  paragraph  in 
Thomaelen-Howe's  Textbook.1 

"The  historical  development  of  the  circle  diagram  is  very  interesting. 
Heyland  published  the  diagram  in  the  E.  T.  Z.  on  the  llth  Oct. 

1894,  and  gave  further  developments  on  pp.  649  and  823  for  the  year 

1895.  In  the  E.  T.  Z.,  1896,  pp.  63,  Behrend  developed  the  diagram 

dans  TElektrotechnische  Zeitschrift,  du  30  Janvier  1896,  p.  63,  ou  se  trouve 
aussi  indiqu6e  pour  la  premiere  fois  la  representation  du  travail  et  du  couple 
moteur. 

"Quant  a  la  representation  du  glissement  par  une  echelle  lineaire,  elle 
a  e"t6  indiquee  pour  la  premiere  fois  dans  mon  article  de  1' Industrie  Electrique, 
25  f  evrier  1896,  ainsi  qu'un  precede  de  correction  due  a  la  resistance  negligee. 

"M.  HEYLAND  a  indique  plus  recemment  une  correction  graphique 
fort  elegante,  mais  qui  parait  peu  rigoureuse,  comme  je  le  montrerai 
prochainement." 

!"A  Textbook  of  Electrical  Engineering,"  by  DR.  A.  THOMAELEN, 
Translated  by  GEORGE  W.  O.  HOWE.  3d  edition,  1912  (Longmans)  p.  386. 


INTRODUCTION  11 

analytically,  but  made  a  small  error  in  the  determination  of  the  rotor 
current.  The  convenient  determination  of  the  slip  and  losses  was  given 
by  Heyland  in  the  E.  T.  Z.  for  1896,  p.  138.  (See  also  Heyland's 
"Eine  Methode  zur  experimentellen  Untersuchung  an  Induktions- 
motoren, '  published  in  Voit's  "Sammlung,"  Vol.  ii,  1900).  Emde 
corrected  Behrend's  error  in  a  letter  to  the  E.  T.  Z.,  1900,  p.  781,  which 
opened  an  interesting  discussion.  In  the  " Z.  fur  E.,"  Vienna,  for  1899, 
Ossanna  gave  the  diagram,  corrected  for  stator  loss.  (See  also  an 
article  by  Ossanna  in  the  E.  T.  Z,,  1900,  p.  712,  and  also  by  Thomaelen 
in  the  E.  T.  Z.,  1903,  p.  972.)  It  is  interesting  to  note,  however,  that 
Ossanna's  circle  was  really  included  in  Heyland's  first  publication." 

References  to  the  circle  diagram  being  the  work  of  A.  Blondel, 
B.  A.  Behrend,  and  A.  Heyland  are  to  be  found  in  the  papers  by 
J.  Bethenod,  L.  Edairage  Electrique,  Aug.,  1904,  and  by  M.  Edou- 
ard  Roth,  of  Belfort,  France,  in  L'Eclairage  Electrique,  Apr.  to 
June,  1909.  The  interesting  small  volume  by  Dr.  K.  Krug1  on 
the  circle  diagram  of  the  induction  motor  contains  the  following 
instructive  historical  reference : 

"The  circle  diagram  for  the  elucidation  of  the  operation  of  induction 
motors,  which  was  published  almost  simultaneously  by  Heyland  and 
Behrend,  took  into  account  only  approximately  the  iron  losses  and 
the  primary  copper  loss.  The  accurate  consideration  of  these  losses 
was  first  given  by  Ossanna  and  his  results  were  developed  later  in  differ- 
ent ways  by  other  authors. 

"In  most  of  these  papers  the  methods  employed  consist  in  a  reduction 
or  adaptation  of  the  Heyland  circle  diagram,  so  as  to  take  account  of  the 
primary  ohmic  drop  as  well  as  of  the  iron  losses  in  accordance  with  the 
facts. 

"Original  proofs  of  the  accurate  circle  diagram  which  may  be  reduced 
to  the  problem  of  the  so-called  general  alternating  current  transformer, 
have  been  given  among  others  by  Lehmann,  by  means  of  vector  analysis, 
by  La  Cour  by  means  of  inversion,  and  by  Petersen  with  the  aid  of  a 
principle  of  superposition. 

"The  following  shows  a  solution  of  the  general  alternating  current 
circuit  by  means  of  complex  algebra." 

Another  point  of  view  is  presented  by  Arnold  and  La  Cour.2 

1  DR.  KARL  KRUG,  "Das  Kreisdiagramm  der  Induktionsmotoren."  Ber- 
lin, J.  SPRINGER,  1909,  p.  5. 

2E.  ARNOLD  and  J.  L.  LA  COUR,  Die  Induktionsmaschinen.  Berlin, 
J.  SPRINGER,  1909,  p.  65.  Also  French  text,  Les  Machines  Asynchrones. 
Premiere  Partie.  Les  Machines  d'Induction.  Paris:  Librairie  Ch.  Dela- 


12  INDUCTION  MOTOR 

"A.  Heyland  showed  first  (E.  T.  Z.,  1895)  that  the  locus  of  the  current 
vector  is  a  circle  for  constant  main  flux,  and  he  gave  a  proof  for  it  in 
E.  T.  Z.,  1896.  Behrend  also  derived  his  relation  from  the  transformer 
diagram,  and  Blondel  has  referred  to  some  relations  in  this  diagram. 
The  diagram  is  sometimes  called  the  Heyland  diagram." 

This  brief  reference  to  the  history  of  the  development  of  the 
theory  is  surely  somewhat  misleading,  as  it  has  been  shown  here 
that  Heyland  supplied  no  proof  of  the  circle  relation  until 
considerably  after  the  publication  of  my  paper  on  Jan.  30,  1896, 
and  then  his  proof  neglected  the  secondary  leakage.  Also,  it 
would  appear  that  the  fundamental  labors  of  A.  Blondel  in  this 
direction  have  been  somewhat  summarily  put  aside  without  the 
recognition  due  them. 

In  the  treatise  of  Kittler  and  Petersen1  we  read  on  p.  486 : 

" Heyland  developed  this  diagram  bearing  his  name.  Through  their 
labors  in  giving  final  form  and  in  clarifying  the  diagram,  Emde  and 
Behrend  have  achieved  preeminent  merit." 

It  is  Interesting  to  note  a  letter  by  Mr.  A.  Heyland,  E.  T.  Z., 
p.  61,  Jan.  21,  1904,  in  which,  after  submitting  a  lengthy 
apology  for  the  use  of  a  single  leakage  field,  he  proceeds  to  say: 
"In  respect  to  the  query  why  I  introduced  at  one  time  the  above 
simplifications  into  the  diagram  (meaning  especially  the  single 
leakage  field),2  allow  me  to  say  that  these  simplifications  (sic!) 
were  thoroughly  warranted  at  the  time.  The  Circle  Diagram  in 
its  more  complex  form  found  little  recognition  9  years  ago  and 
remained  almost  unknown.  (The  italics  are  my  own.)  It 

grave,  1912,  p.  64.  "  A.  HEYLAND  a  signale  le  premier  (E.  T.  Z.,  1895)  que 
le  lieu  de  1'extremite  du  vecteur  du  courant  etait  un  cercle  si  le  flux  principal 
est  maintenu  constant;  il  en  a  donne  une  explication  en  1896  dans  la  E.  T.  Z., 
BEHREND  a  egalement  deduit  cette  the"orie  (E.  T.  Z.,  1896);  il  1'a  tiree  du 
diagramme  du  transformateur;  BLONDEL  en  a  deduit  a  son  tour  quelques 
relations.  On  appelle  souvent  ce  diagramme  le  diagramme  D'HEYLAND." 

^'Allgemeine  Elektrotechnik."  Edited  by  Dr.  E.  KITTLER.  Vol.  II, 
"Einfiihrung  in  die  Wechselstromtechnik."  By  W.  PETERSEN.  Stutt- 
gart: F.  ENKE,  1909.  "EMDE  und  BEHREND  haben  sich  durch  ihre  Arbeiten 
um  die  endgtiltige  Formgebung  und  Klaerung  des  Diagram  mes  in  hervorra- 
gender  Weise  verdient  gemacht." 

2  "A  Graphical  Treatment  of  the  Induction  Motor."  By  ALEXANDER 
HEYLAND.  Translated  by  G.  H.  ROWE  and  R.  E.  HELLMUND.  McGraw 
Pub.  Co.,  New  York,  1906.  The  entire  paper  seems  affected  by  this 
assumption. 


INTRODUCTION  13 

became  known  only  after  the  publication  of  the  simplified 
construction  of  the  circle,  which  I  published  in  a  letter  to  the 
E.  T.  Z.,  1895,  p.  649,  which  represented  an  excerpt  of  a  paper  in 
The  Electrician  in  Feb.  14,  1896." 

I  think  Mr.  Heyland  is  right  that  his  paper  of  Oct.  1894 
"remained  almost  unknown."  I  think  I  agree  with  Mr.  Heyland 
that  it  was  necessary  to  give  a  simple  demonstration  of  the  theory 
and  a  simple  geometrical  proof  in  order  to  introduce  the  Circle 
Diagram  to  the  engineer.  This  simple  demonstration  and 
geometrical  proof  were  first  given  by  me,  and  Mr.  Heyland 's 
prior  and  later  publications  in  no  wise  detract  from  this  fact. 
When  Mr.  Heyland  saw  that  I  had  succeeded  in  giving  a  treat- 
ment which  was  as  accurate  as  his  own  paper  in  1894  but  a  great 
deal  simpler,  and  in  which  nothing  was  neglected  which  he 
there  took  into  account,  excepting  the  primary  resistance,  he 
endeavored  to  obtain  an  equally  simple  method  of  treatment  and 
he  tried  to  prove  that  the  secondary  leakage  coefficient  was  non- 
existent1 and  that  that  which  had  been  viewed  as  secondary 
leakage  was  merely  part  of  the  primary  leakage  and  in  time 
phase  with  the  primary  current.  Then  he  introdued  one  of 
the  most  unhappy  errors  which,  due  to  his  authority,  has  not 
yet  completely  vanished.  He  also  introduced  the  circular  arcs 
for  the  representation  of  the  copper  losses  in  the  primary  and 
secondary  windings,  which  were  superseded  a  few  years  later  by 
the  straight  lines  as  given  in  Fig.  5  reproduced  from  Fig.  56,  p. 
101,  of  the  1st  edition  of  the  present  author's  book,  "The 
Induction  Motor,"  1901. 

We  have  reproduced  as  Fig.  4,  Figs.  1,  2  and  3  of  Mr. 
Heyland's  paper,  p.  561,  E.  T.  Z.,  Oct.  11,  1894;  as  Figs. 
6  and  7,  Figs.  2  and  3  of  my  own  paper,  pp.  63  and  64,  E.  T.  Z., 
Jan.  30,  1896;  and  as  Figs.  8  and  9,  Figs.  226  and  227,  pp.  224 
and  228  of  Silvanus  P.  Thompson's  "Polyphase  Electric  Cur- 
rents," 2d  edition,  1900.  We  suggest  to  the  reader  a  careful 
study  of  these  figures  and  we  may  then  leave  it  safely  to  his 
judgment  whether  I  utilized  in  my  early  paper  any  of  Mr.  Hey- 
land's work,  or  whether  Mr.  Heyland  and  the  late  Prof.  S.  P. 

1  F.  EMDE,  E.  T.  Z.,  p.  855,  1900,  where  we  also  read:  "On  this  occasion  I 
wish  to  refer  to  HEYLAND'S  paper,  No.  41,  E.  T.  Z.,  1894,  which  at  any  rate 
excels  in  accuracy  his  later  papers,  though  it  lacks  the  'seductive'  simplic- 
ity and  it  is  therefore  referred  to  only  historically."  EMDE  has  been  one  of 
HEYLAND'S  strongest  admirers,  and  surely  one  of  the  ablest. 


14 


INDUCTION  MOTOR 


Thompson  used  my  early  paper  in  their  own  work  which  came 
out  after  the  appearance  of  my  paper.    It  is  true  that  Prof.  S.  P. 


90*  Slip 


C, 


Generator 


Motor 


FIG.  5. — Facsimile  of  Fig.  56,  p.  101,  of  the  First  Edition  of  B.  A.  Behrend's 
book,  "The  Induction  Motor,"  New  York,  McGraw  Publishing  Co.,  1901. 

Thompson  cited  my  paper  in  the  bibliography  in  "Polyphase 
Electric  Currents;"  it  is  also  true  that  he  did  me  the  honor  of 


INTRODUCTION 


15 


FIG.  6. — Facsimile  of  Fig.  2  of  B.  A.  Behrend's  paper  "On  the  Theory  of  the 
Polyphase  Motor,"  Jan.  30,  1896,  E.  T.  Z. 


FIG.  7. — Facsimile  of  Fig.  3  of  B.  A.  Behrend's  paper  "On  the  Theory  of  the 
Polyphase  Motor,"  Jan.  30,  1896,  E.  T.  Z. 


16 


INDUCTION  MOTOR 


naming1  my  little  treatise  " The  Induction  Motor"  as  one  of  the 
three  books  on  the  theory  of  alternating-current  motors  which 
he  recommended  to  his  readers,  and  I  would  fain  refrain  from 
showing  how  much  my  work  has  been  used  by  these  authors  if 
it  were  not  for  the  fact  that  during  the  last  25  years  I  have 
remained  silent,  trusting  to  the  fairness  of  authors  not  to  deprive 
a  fellow  author  of  the  just  credit  due  him. 


FIG.  8. — Facsimile  of  Fig.  226,  p.  224,  of  Silvanus  P.  Thompson's  book,  "Poly- 
phase Electric  Currents,"  2d  Edition,  1900. 

A  typical  case  in  question  is  the  reference  on  p.  413  in  my 
friend  Miles  Walker's  "  Specification  and  Design  of  Dynamo- 

1  SILVANUS  P.  THOMPSON,  "Dynamo-Electric  Machinery,"  7th  ed., 
Vol.  i,  p.  36,  London,  E.  &  F.  N.  Spon,  Ltd.  1904.  "The  theory  of 
alternate  current  motors  of  the  asynchronous  and  of  the  synchronous  types 
has  of  late  received  much  attention  from  various  writers.  The  reader  is 
referred  to  the  author's  book  "Polyphase  Electric  Currents"  (2d  ed.,  1900) 
to  STEINMETZ'S  "Alternating  Current  Phenomena"  (3d  ed.,  1900),  or  to 
BEHREND'S  "The  Induction  Motor"  (1901). 


INTRODUCTION 


17 


Electric  Machinery,"  Longmans,  Green  &  Co.,  1915,  which 
shows  the  circle  diagram  as  given  on  p.  101  of  "The  Induction 
Motor,"  New  York,  1901,  but  with  the  following  comment: 

"It  is,  therefore,  convenient  to  reproduce  here  a  form  which  is  found 
to  be  very  convenient  in  workshop  use,  and  to  give  results  which  check 
sufficiently  with  those  obtained  in  practice." 


FIG.  9. — Facsimile  of  Fig.  227,  p.  228,  of  Silvanus  P.  Thompson's  book,  "Poly- 
phase Electric  Currents,"  2d  Edition,  1900. 

Reference  is  made  to  a  footnote  which  begins: 

"See  KarapetofFs  'Experimental  Electrical  Engineering/  Vol.  ii, 
p.  166;  Cramp  and  Smith,  'Vector  Diagrams;'  Graphical  Treatment 
of  the  Rotating  Field,  R.  E.  Hellmund,  A.  I.  E.  E.  Proceedings,  p.  927, 
1918,  etc.,  etc.  .  ." 

The  present  form  of  the  circle  diagram,  as  applied  to  the  solu- 
tion of  induction  motor  and  transformer  problems,  and  the 
methods  of  its  demonstration  and  proof,  are  based  upon  BlondeFs 
diagram  of  the  composition  of  fluxes  and  upon  the  proof  of  the 
circular  locus  as  developed  by  the  present  author.  Mr.  Hey- 
land's  contributions  to  the  subject,  however  interesting  and 
suggestive  they  have  been,  have  not  survived. 


18  INDUCTION  MOTOR 


THE   DEVELOPMENT  OF  THE  THEORY  OF  THE  SINGLE-PHASE 
INDUCTION  MOTOR 

The  analytical  theory  of  the  single-phase  induction  motor  owes 
much  to  the  labors  of  Potier,  Dr.  Behn-Eschenburg,  Goerges, 
Steinmetz,  and  McAllister.  As  the  analytical  theory  has  always 
been  somewhat  abstruse,  an  attempt  was  made  by  the  author  as 
early  as  1896  to  represent  the  locus  of  the  primary  current 
through  graphical  analysis,  and  it  was  found  that  the  primary 
current  in  the  polar  diagram  could  be  represented  by  vectors 
drawn  from  a  pole  to  the  circumference  of  a  circle.  This  was 
proved,  however,  only  for  a  limited  case,  viz.  for  a  motor  in 
which  the  secondary  resistance  was  partially  negligible. 

This  analysis  of  the  operation  of  the  single-phase  induction 
motor  by  means  of  a  proof  that  the  primary  current  locus  is  also 
a  circle,  was  given  by  the  author  in  the  E.  T.  Z.,  March  25,  1897. 
The  analysis  was  carried  through  by  dissolving  the  single  oscil- 
lating field  into  two  equal  and  oppositely  rotating  fields.  It  was 
assumed  that  the  rotor  resistance  of  the  second  motor  with 
reverse  torque  was  negligible.  With  these  assumptions  a  circle 
represents  correctly  the  locus  of  the  primary  current. 

The  same  analysis  was  repeated  in  the  first  edition  of  the 
author's  book  on  "The  Induction  Motor." 

Utilizing  the  able  papers  of  H.  Goerges  in  the  E.  T.  Z.,  1895 
and  1903,  on  the  single-phase  induction  motor,  in  which  Goerges 
introduced  the  cross  field,  Prof.  J.  K.  Sumec  gave  a  comprehen- 
sive and  elegant  graphical  solution  which  remains  perfectly  simple 
in  spite  of  its  accuracy.  His  first  paper  was  published  in  the 
"Zeitschrift  fur  Elektrotechnik,"  Vienna,  No.  36,  1903.  The  sub- 
ject is  most  admirably  treated  in  a  little  pamphlet  entitled  "Der 
einphasige  Induktionsmotor,"  by  J.  K.  Sumec,  Nov.  20,  1904, 
reprinted  from  the  "Archiv  der  Mathematik  und  Physik," 
Leipzig,  B.  G.  Teubner. 

Treating  the  theory  of  the  single-phase  induction  motor  by 
means  of  a  resolution  of  the  oscillating  field  into  two  equal 
oppositely  rotating  fields,  Dr.  A.  Thomaelen,  in  E.  T.  Z.,  1905, 
p.  1,111  et  seq.,  arrives  at  the  same  result  as  that  given  by  Sumec 
without  neglecting  the  rotor  resistance  which  was  the  new  ele- 
ment in  Sumec's  work.  Dr.  Thomaelen's  treatment,  however, 
is  rather  complex  and  its  value  consists  in  proving  that  the  two 
methods  lead  to  the  same  result. 


INTRODUCTION  19 

This  has  again  been  proved  by  Arnold  and  LaCour  in  "Les 
Machines  d'Induction,"  Part  I,  p.  149,  of  the  French  edition, 
Paris,  Ch.  Delagrave.  The  authors  have  used  the  method  of 
equivalent  circuits  which  they  have  employed  throughout  their 
work.  It  may,  therefore,  be  safely  assumed  that  both  methods 
of  analysis  give  identical  results. 

Reference  must  here  be  made  to  the  seventh  edition,  1918,  of 
Dr.  A.  Thomaelen,  "Kurzes  Lehrbuch  der  Elektrotechnik," 
which  has  just  come  to  our  attention.  Throughout  the  treat- 
ment of  the  theory  of  the  induction  motor,  both  poly-phase  and 
single-phase,  Dr.  Thomaelen  has  used  the  author's  leakage  coef- 
ficients, assuming  apparently  that  they  are  novel  and  expressing 
his  satisfaction  that  they  give  results  easily  and  clearly.  Since 
the  present  author  introduced  these  coefficients  in  his  first  mono- 
graph of  1896  and  as  they  have  been  used  since  with  full 
credit  by  Messrs.  Kapp  and  Sumec,  he  likes  to  point  out  again 
that  in  this  work  he  is  using  the  reciprocals  of  the  coefficients 
which  he  used  in  his  early  monographs  and  in  the  first  edition 
of  this  book.  Thus,  the  coefficients  are  the  same  as  those  of 
Hopkinson,  as  they  were  adopted  in  1894  by  our  great  master, 
Andre  Blondel. 


CHAPTER  II 
THE  THEORY  OF  FLUXES  AND  STRAY  FIELDS 

The  problem  of  problems,  in  the  solution  of  which  the  elec- 
trical engineer  is  deeply  interested,  and  which  underlies  all  others, 
is  set  before  us  in  the  form  of  the  alternating-current  transformer 
possessing  considerable  leakage  and  a  relatively  large  magnetizing 
current. 

A  choking  coil  of  n  turns  or  a  transformer  with  an  open  secon- 
dary, takes  from  the  primary  mains  just  so  much  current  as  is 
necessary  to  produce  a  magnetic  field  F,  which  balances  the  pri- 
mary voltage  ei.  The  induced  voltage  e,  opposite  in  time-phase 
to  the  impressed  voltage  e\  is 

e  =  -n^-10-8  volts  (1) 

The  magnetizing  current — neglecting  for  the  moment  hysteresis 
and  eddy  currents — lags  behind  the  primary-impressed  voltage 
by  a  quarter  of  a  time-phase.  It  leads  by  a  quarter  of  a  time- 
phase  over  the  induced  counter  e.m.f.  We  say,  "  It  is  in  quadra- 
ture with  the  impressed  e.m.f."  The  product  of  this  current 
into  the  impressed  e.m.f.,  integrated  over  the  time  of  one  com- 
plete period,  i.e.,  the  work  done  by  this  current,  is  zero.  Currents 
in  quadrature  with  the  e.m.f.  have  been  called  by  M.  Dobrowolsky 
"  watt-less"  currents. 

Any  transformer,  induction  motor,  or  other  alternating-cur- 
rent device  of  any  sort  or  description,  under  load  or  under  no 
load,  has  one  and  only  one  primary  magnetic  field  resulting  from 
the  actions,  and  interactions,  of  its  current-carrying  coils.  The 
magnitude  of  this  resultant  magnetic  field  is  such  that  its  varia- 
tion produces  a  counter  e.m.f.  in  phase  with  the  impressed  e.m.f., 
and  of  such  magnitude  as  will  permit  the  flow  of  the  primary 
current  through  the  ohmic  resistance  of  the  primary  coils. 

The  fundamental  importance  of  this  statement  must  be  em- 
phasized as  it  is  applied  throughout  this  book. 

A  choking  coil  of  sectional  area  A,  of  magnetic  reluctance  p, 
of  ohmic  resistance  r,  and  number  of  terms  n,  placed  in  a  circuit 

20 


THE  THEORY  OF  FLUXES  AND  STRAY  FIELDS          21 

of   frequency  ~,  with  effective   (square  root  of  mean  square) 
current  i,  carries  a  maximum  flux  F,  of  maximum  induction  B : 

Fmax  =  A  •  Bmax  (2) 

Hds 


p 
J*Hds  is  the  line-integral  of  the  magnetic  force  H 


I 


Hds  =  0.47T/  (4) 


which,  for  any  closed  circuit,  is  equal  to  0.4rr  times  the  entire 
magnetizing  ampere-conductors.  If  the  integral  taken  around  a 
closed  circuit,  viz.  the  "magneto-motive  force,"  is  zero,  no  flux 
can  result  from  the  currents  around  which  the  integral  was  taken. 
A  neglect  of  this  fundamental  conception  of  the  theory  of  electro- 
magnetism  has  led  to  false  diagrams  of  stray-fields. 
Therefore, 


R  _     .  /_>. 

— 

We  assume  the  time  variation  of  the  flux  to  follow  a  simple  sine 
law 

F  =  Fmax  •  sin  co*  (6) 

0,    =     27T~  (7) 

Therefore  from  (1) 

et=  -rcJ-H)-8  (1) 

=   —  co.Fmox  •  cos  co/  •  10~8 

=  -2?r  ~  Fmax  cos  co*  .  10~8  (8) 

emax  =  e\/2  (9) 

/.  e  =  -4A4~nFmax  •  lO"8  (10) 

From  (8)  it  is  apparent  that  e,  the  induced  e.m.f.,  lags  90  time 
degrees  behind  the  inducing  current.  The  impressed  e.m.f.  there- 
fore leads  the  current  by  90  time  degrees.  (See  Fig.  10.) 

All  our  polar  diagrams  rotate  in  the  positive  direction,  which 
is  counter-clockwise  by  international  agreement. 

The  ohmic  drop  requires  the  addition  of  ir  in  time-phase  with 
the  flux  to  the  impressed  voltage  d,  requiring  EI  as  the  final 
resultant  impressed  voltage. 

The  placing  of  a  secondary  coil  on  the  magnetic  circuit 
makes  the  device  a  transformer. 


22 


INDUCTION  MOTOR 


If  we  assume  that  both  primary  and  secondary  coils  embrace 
the  entire  flux,  there  being  no  stray  or  leakage  fluxes,  and  if  we 
assume  the  number  of  turns  to  be  the  same  in  both  circuits,  then 


ma* 


FIG.  10. 


FIG.  11. 


61  and  62  are  the  e.m.fs.  impressed  upon  both  the  primary  and 
secondary  circuits,  respectively.  If  the  load  on  the  secondary 
circuit  is  non-inductive,  the  current  i2  is  in  phase  with  e.  The 
primary  current  must  be  such,  in  phase-direc- 
tion and  magnitude,  that  the  resultant  m.m.f. 
of  the  primary  and  secondary  ampere-turns 
produces  the  field  F  in  magnitude  and  time- 
phase. 

Knowing  e\,  we  find  F',  knowing  p  we  find  the 
primary  ampereturns  for  magnetization,  rep- 
resented by  i^  in  Fig.  11.  Adding  itfi  to  e\ 
gives  El,  the  resultant  impressed  voltage. 

There  is  another,  and  a  safer  way,  proposed 
by   Prof.   Andre  Blondel   in  a  famous  paper 
__  entitled,   "Quelques    proprietes   generates  des 

quantities    in    the    champs    magnetiques    tournants"    (Eclairage 


12N°     ElectriQUe>    10>    17>    24    AuS->     1895)>    in    which 

the  magnetic  fluxes  are  composed  as  follows: 
If  acting  alone  i%  would  produce  a  flux  4>2  equal  to  X2  -5-  p,  where 
Xz  represents  the  m.m.f.  of  2*2,  and  p  the  reluctance  of  the  mag- 
netic circuit  in  common  to  both  primary  and  secondary  circuits; 
if  acting  alone  i\  would  produce  a  flux  <J>i  equal  to  Xi  -=-  p, 
where  Xi  represents  the  m.m.f.  of  i\,  and  p  again  the  reluctance 


fa*. — 


(Facing  page  22) 


THE  THEORY  OF  FLUXES  AND  STRAY  FIELDS  23 

of  the  magnetic  circuit  in  common  to  both  primary  and  secondary 
circuits.  3>2  vectorially  subtracted  from  3>i  must  leave  F  in  mag- 
nitude and  direction  (Fig.  12). 

The  great  advantage  of  this  method  becomes  apparent  in  the 
treatment  of  the  theory  of  the  transformer  with  leakage  and  in  the 
more  complex  problems  of  double  squirrel-cage  motors,  con- 
catenation, etc.  Its  disadvantage  lies  in  the  danger  of  looking 
upon  the  fictitious  3>i  and  $2  as  fluxes  actually  in  existence 
and  having  physical  entity.  We  shall  use  both  methods  wherever 
they  represent  closely  the  physical  phenomena. 

It  is  well  known  in  dynamo  design,  as  first  taught  us  by  Dr. 
John  Hopkinson,  that  the  flux  threading  the  primary  does  not 
reach  the  secondary  without  leakage  or  stray  fields.  If  we  as- 
sume with  Prof.  A.  Blondel  that  the  ratio  of  the  flux  of  the  pri- 
mary to  that  which  reaches  the  secondary  is  vi,  where  v\  is  greater 
than  1,  and  the  ratio  of  the  secondary  flux  to  that  which  reaches 
the  primary  is  t>2,  where  v2  is  also  greater  than  1,  then  (vi  —  l)3>i 
and  (v2  —  1)$2  represent  the  stray  fluxes  or  leakage  fluxes,  which 
are  in  time-phase  with  their  respective  m.m.fs.  or  currents. 

In  my  paper  E.  T.  Z.,  Jan.  30,  1896,  and  in  the  first  edition  of 
this  book,  I  used  the  reciprocals  of  BlondeFs  v's.  Though  un- 
fortunately most  authors  have  since  followed  my  use  of  these 
coefficients,  as  Silvanus  P.  Thompson,  Gisbert  Kapp,  Alexander 
Gray,  J.  K.  Sumec,  A.  Thomaelen,  and  others,  after  very  careful 
consideration,  I  have  become  convinced  that  it  is  better,  in  the 
interest  of  uniformity  and  clearness,  to  give  up  my  coefficients, 
which  were  smaller  than  1,  and  instead  to  adopt  BlondePs,  which 
are  larger  than  1,  and  this  practice  also  conforms  to  the  disper- 
sion coefficients  of  Dr.  Hopkinson's  which  are  also  greater  than  1. 
This  matter  is  solely  a  convention  and  in  no  manner  affects  the 
accuracy  or  correctness  of  our  arguments  or  of  previous  papers. 
There  is  much  to  be  said  for  the  retention  of  my  old  coefficients  as 
they  are  logical  in  viewing  the  deviation  of  the  ratio  of  trans- 
formation at  no  load  from  the  ideal  ratio  as  the  measure  of  the 
leakage.  In  my  early  notation,  the  primary  and  secondary  leak- 
age fields  were: 


Behrend's  old  Notation  (11) 

/-•->)*• 


24  INDUCTION  MOTOR 

In  B  lenders  notation: 


/2    =    (02    ~ 


Blondel' s  Notation 


(12) 


The  utmost  care  is  essential  to  avoid  confusion  and  I  believe  a 
service  is  rendered  by  the  adoption  of  a  uniform  notation. 

The  diagram  of  fluxes  can  now  be  drawn  directly  (Fig.  13  and 
14). 


FIG.  13. — The  flux  diagram  of  the 
induction  motor  or  transformer,  in- 
cluding leakage. 


FIG.  14. — Electric  circuits  sim- 
ulating the  leakage  paths  of  the 
magnetic  circuit  of  the  induction 
motor. 


F2  induces  ez  + 

fa  =  Xz  -r-  p2  =  (#2  ~-  1)^2  secondary  leakage  flux 

fi  =  Xi  -T-  pi  =  (vi  —  l)3>i  primary  leakage  flux 

FI  —  resultant  primary  flux 

3>2  =  Xz  -T-  p  fictitious  secondary  flux 

$1  =  Xi  -4-  p  fictitious  primary  flux 

Ei  =  ei  -f-  iiri  primary  impressed  voltage 

It  is  very  desirable  to  keep  in  mind  a  picture  of  the  corre- 
sponding electric  currents  with  their  e.m.fs.  and  distribution  of 
resistances.  In  Fig.  14,  Xi  and  X2  represent  the  primary  and 


THE  THEORY  OF  FLUXES  AND  STRAY  FIELDS 


25 


secondary  m.m.fs.,  p,  pi,  P2,  the  reluctances  of  the  common  and 
leakage  paths.  The  fluxes  are  entered  and  the  diagram  shows 
clearly  how  the  leakage  fluxes  become  cumulative  by  vectorial 
addition. 

Figures  13  and  14  should  always  be  kept  together  before  the 
mind,  with  the  underlying  assumption  that  the  reluctance  of 
the  iron  is  assumed  as  negligible,  in  fact  zero. 


FIG.   15. — The  polar  diagram  for  constant  current. 

Mr.  G.  Kapp  originated  a  method  before  the  advent  of  the 
Blondel  flux  diagram,  which  is  still  adhered  to  by  Dr.  Steinmetz, 
in  which  the  e.m.fs.  induced  by  the  leakage  fields  /i  and  /2  are 
represented  lagging  by  a  quarter  phase  the  primary  and  secon- 
dary currents.  We  refer  to  the  author's  Fig.  6  from  his  original 
paper  of  Jan.  30,  1896,  in  which  both  methods  are  shown  in  the 
diagram,  and  from  which  it  is  apparent  that  the  flux  method  of 
Blondel  is  both  more  nearly  in  keeping  with  the  physical  facts 
and  a  great  deal  simpler  in  its  geometrical  interpretation.  The 
results  obtained  by  both  methods  are,  of  course,  identical,  though 


26  INDUCTION  MOTOR 

there  seems  now  little  warrant  for  retaining  the  older  method  of 
Mr.  Kapp's  as  done  throughout  in  the  works  of  Dr.  Steinmetz 
and  in  the  recent  textbook  of  Prof.  R.  R.  Lawrence,  "  Principles 
of  Alternating  Current  Machinery,"  McGraw  Hill  Book  Co., 
1916.  At  least  the  flux  method  should  be  considered  beside  the 
older  conventional  one. 

We  are  still  concerned  with  the  transformer.  We  wish  to 
know  how  the  magnitude  and  phase  of  the  primary  e.m.f.  vary 
with  constant  primary  current  and  varied  secondary  resistance. 
As  Oe  is  proportional  to  the  primary  current,  Fig.  15,  we  shall 
assume  it  to  remain  constant,  neglecting  for  the  present  the 
primary  resistance  which  is  easy  to  take  into  account.  The  angle 
Oaw  is  a  right  angle,  hence,  describe  a  semi-circle  over  Oe  as 
diameter,  then  by  varying  the  secondary  resistance  we  vary  Fz  = 
Oa,  which  is  in  quadrature  with  and  proportional  to  ez,  the  secon- 
dary voltage.  Remember  that  to  obtain  the  secondary  terminal 
voltage  we  must  deduct  i2r2  the  ohmic  drop  in  the  transformer 
windings  from  e2.  As  Od  =  Oc  -r-  v\y  it  follows  that  Od  is  a 
measure  of  the  primary  voltage.  The  point  d  divides  ae  in  the 
same  ratio  for  all  configurations  of  the  diagram,  as  is  easily 
shown. 

ab  =  (v2  — 

bd  =  be  —  ed 


/.  ab  +  bd  =  (v2  -  -} 

\          v\i 


ad  =  ab  +  bd  =  (viV2  —  1)  - 

v\ 


.'.  ad  -r-  ed  =  ViV2  —  1  (13) 

a  =  ViV2  —  I  (14) 

We  shall  call  ViV2  —  1  by  the  Greek  letter  0,  which  we  shall  see 
later  is  the  most  characteristic  constant  of  a  transformer  or 
induction  motor  and  it  is  usually  called  the  Leakage  Factor.  [In 
my  former  notation,  my  old  coefficients  being  the  reciprocals  of 
the  Blondel  coefficients  here  used,  the  Leakage  Factor  was  equal 

to  a  =  ---  1.1     Describing  now  the  semi-circle  edk  so  that 


Ok  -T-  ke  =  <r  =  ViV2  —  1,  then  draw  ae  at  its  intersection  with 
the  semi-circle  edk,  at  d  we  obtain  the  polar  ray  Od,  which  is 
1  -r  Vi  times  the  primary  field  Fi. 


THE  THEORY  OF  FLUXES  AND  STRAY  FIELDS 


27 


Taking  now  into  account  the  primary  resistance,  we  consider 
that  iiri  is  produced  by  a  field  proportional  to  itfi  and  in  quadra- 
ture with  its  time-phase.  If  Oc  measures  61,  to  an  arbitrary  scale, 
then  ch  measures  to  the  same  scale  itfij  and  as  ii  is  constant 
in  a  constant  current  transformer  (primary  current  constant), 
iiTi  is  constant  also,  and  therefore,  ch  constant,  which  is  the  field 
corresponding  to  it.  Draw  Oh,  then  ddr  parallel  to  ch,  and  so  on 
for  any  number  of  points  "d."  Obviously,  the  point  d'  lies  on 
the  semi-circle  e'd'k' ,  whose  center  0\  is  determined  by  the  dis- 
tance O'Oi'  =  ddf. 


FIG.   16. — The  polar  diagram  for  constant  voltage. 

The  reader  should  plot  from  this  simple  polar  diagram  in 
Cartesian  coordinates  the  primary  e.m.f.,  the  cos  <p,  and  the  sec- 
ondary current,  relations  ordinarily  obtained  by  long  formulas. 
Of  course,  if  we  like  these  long  formulas,  we  can  have  them  by 
reading  them  off  the  diagram  in  complex  algebra  or  in  trigon- 
metric  functions.  He  should  see  what  is  evident  by  inspection 
of  the  diagram — that  large  leakage  will  make  the  transformer 
into  a  choking  coil. 

What  happens  if  the  primary  voltage  is  kept  constant  instead 
of  the  primary  current?  The  flux  diagram  yields  a  direct  and 
beautifully  simple  answer  (Fig.  16).  Neglecting  for  the  present 
the  primary  resistance,  F\  is  constant  with  constant  impressed 
e.m.f.,  therefore,  Od  is  constant,  if  the  secondary  resistance  is 
varied  in  any  manner.  We  found  above  in  equation  (13)  that 


28 


INDUCTION  MOTOR 


ad  -t-  ed  =  ViV2  —  1,  and  as  Z  Oae  is  a  right  angle,  draw  em  and 
triangles  A  Oad  ~  A  med,  and 


Od  -T-  dm 


—  1 


(15) 


Hence,  the  point  e  describes  a  semi-circle  over  dm  as  diameter. 
This  is  the  identical  proof  given  by  the  author  in  his  paper  Jan. 
30,  1896,  and  the  first  proof  ever  given  of  this  remarkable  relation. 


o  d 

No  Leakage  Leakage 

FIG.  17. — Transformer  with  and  without  leakage. 

The  effect  of  the  leakage  fields  consists  in  turning  the  polar  ray 
of  the  primary  current  from  its  locus  on  the  line  dy  which  it 
would  have  without  leakage,  counter-clockwise  upon  the  semi- 
circle into  which  the  line  dy  has  been  transformed.  The  phase 
lag  is  therefore  increased  and  the  power  factor  diminished  by  the 
presence  of  leakage  in  the  transformer.  Figure  17  shows  this 
once  more. 


CHAPTER  III 
THE  GENERAL  ALTERNATING-CURRENT  TRANSFORMER 

A.  THE  TRANSFORMER  WITH  NON-INDUCTIVE  LOAD 

In  an  alternating-current  transformer  with  resistance  and 
leakage,  a  resultant  primary  field  FI,  which  is  the  real  field  and 
which  is  produced  by  the  m.m.f  .  resulting  from  the  interaction  of 
the  m.m.fs.  of  the  primary  and  secondary  windings,  induces  in 
the  primary  windings  a  counter  e.m.f.  which,  added  vectorially 
to  the  impressed  e.m.f.,  leaves  a  resultant  e.m.f.  which  is  equal 
to  the  ohmic  drop  in  the  primary  windings. 

If  the  secondary  of  the  transformer  is  open  we  are  led  to  the 
simple  classic  statement  due  to  G.  Kapp  that, 

11  A  transformer  working  on  open  secondary  circuit  must  take  from 
the  primary  mains  sufficient  current  to  produce  that  field  which  will 
just  balance  the  primary  voltage.  This  current  is  called  the  open  circuit 
current."1 

The  component  of  the  open  circuit  current  in  phase  with  the 
e.m.f.  supplies  the  losses  incident  to  the  ohmic  loss  in  the  pri- 
mary circuits,  eddy  currents,  and  hysteresis,  and  has  therefore 
aptly  been  called  by  M.  Dolivo-Dobrowolsky  the  "watt-compo- 
nent," in  contradistinction  to  the  "  watt-less"  component,  which 
is  in  quadrature  with  the  e.m.f.  and  which  magnetizes  the  core, 
and  lags  behind  the  impressed  e.m.f.  by  a  quarter-time  period. 
As  seen  in  Chap.  II,  Eq.  (10)  the  impressed  e.m.f.  e\  ("effective," 
or  "  square  root  of  mean  square")  is: 


61  =  4M~nFmaxlO~8  volts  (16) 

where  ~  is  the  primary  frequency,  n  the  number  of  complete 
turns  around  the  core,  and  Fmax  =  F\  the  maximum  value  of  the 
flux. 

The  secondary  current  i*  and  the  secondary  impressed  e.m.f. 

1GiSBERT  KAPP,   "Dynamos,   Alternators,  and  Transformers,"  p.  443. 
London:  Biggs  &  Co.,  139-140  Salisbury  Court,  Fleet  Street,  E.  C.,  1893. 

29 


30 


INDUCTION  MOTOR 


e-2,  which  is  also  the  induced  e.m.f.  of  the  secondary  circuit,  will 
now  be  assumed  to  be  in  time-phase,  in  other  words  the  trans- 
former is  closed  upon  a  non-inductive  resistance  #2.  Then 

e2  =  izR*  (17) 

In  order  to  obtain  a  clear  diagram  we  assume  the  leakage  fluxes 
(t>i  —  1)  and  (z;2  —  1)  to  be  large,  say,  vi  =  1.04  and  vz  =  1.06, 
then  the  leakage  factor  a  becomes  (Chap.  II,  Eq.  (14)), 

-  1  (14) 


<7    = 

a-  =  0.1 


=  .834 
FIG.   18. — The  circle  diagram  for  constant  voltage. 


Assume,  FI  =  12,  and  d  =  120  volts 
D  =  -1  =  120 

<7 


(18) 


With  these  values  we  construct  the  polar  diagram,  Fig.  18, 
whose  values  are  shown  in  Cartesian  coordinates  in  Fig.  19. 
The  maximum  power  factor  obtainable  is 


cos  \f/Q  = 


OiPi 
OiO 


_ 
2(7 


(19) 


THE  GENERAL  ALTERNATING-CURRENT  TRANSFORMER    31 


This  simple  relation  gives  at  a  glance  the  highest  possible  power 
factor  for  a  given  amount  of  leakage  for  non-inductive  load.  For 
our  numerical  case,  we  have 

1 

cos  ^ 


1.2 


=  0.834 


K.W.Inp,ir-  9  87 

FIG.  19.  —  Characteristic  curves  of  transformer  in  Cartesian  coordinates. 

THE  AUTHOR'S  METHOD  OF  ACCOUNTING  FOR  PRIMARY 
RESISTANCE 

The  primary  resistance  adds  to  the  impressed  e.m.f.  e\  the 
component  i\r\  in  phase  with  i\.  Before  we  proceed  further  it 
may  be  advisable  to  call  attention  to  the  fact  that,  in  going  from 
the  flux  diagram  to  the  current  diagram,  the  primary  current  is 
proportional  to  Oe,  the  secondary  current  to  be,  and  the  resultant 
magnetization  to  Oe.  In  Fig.  16,  therefore,  if  Oe  is  drawn  to 


but  —  ,  while  Od  represents  the 


represent  ii,  ed  represents,  not 

open  circuit  current  (neglecting  losses)  which  produces  the  total 
primary  flux  Oc  =  F\. 


32 


INDUCTION  MOTOR 


Instead  of  adding  zVi  to  e\,  and  obtaining  EI,  and  then  turning 
the  current  vector  Oe  =  i\  in  the  positive  or  counter-clockwise 
direction  so  that  EI  coincides  again  in  phase  with  ei}  apply  the 
following  simple  geometrical  device  (Fig.  20). 

Make  triangle  Oeg  similar  to  triangle  Ohk,  then 

eg\0e  ::hk  :  Oh 
i\ .  ' 


eg 


(20) 


FIG.  20. — The  author's  method  of  taking  into  account  the  primary  resistance. 


The  new  current  iY  must  be  in  the  time-phase  Og  and  its  mag- 
nitude remains  equal  to  On  =  Oe  if  the  impressed  e.m.f .  is  raised 
from  ei  to  EI. 

If,  however,  we  assume  the  impressed  e.m.f.  EI  to  be  reduced 
to  Op,  the  value  of  et,  then  the  current  Oe  =  ii  is  to  be  reduced 
in  the  ratio  Og  -5-  Oh  =  e\  -4-  EI,  or  Or  -4-  Oe.  The  magnitude 
of  the  real  primary  current  is  therefore  represented  by  Or  and 
its  phase  by  Og,  its  magnitude  and  phase,  therefore,  by  Os. 

The  method  here  described  is  rapid  and  easy.     It  is  carried 


THE  GENERAL  ALTERNATING-CURRENT  TRANSFORMER    33 

out  in  the  figure.     Its  advantage  lies  in  its  convenient  and  ready 
application  to  all  sorts  of  alternating-current  problems. 

ANOTHER  METHOD  OF  ACCOUNTING  FOR  PRIMARY  RESISTANCE 

A  simple  and  elegant  geometrical  method  applicable  to  the  circle 
diagram  has  been  given  by  Prof.  J.  Sumec,1  of  the  Czecho-Slovac 
University  of  Brtinn,  which  we  shall  now  proceed  to  explain. 


AP  .  AR  =  OA2 
AP  =  As  I  — 


FIG.  21. 

As  .  Ar  =  Ab2  =  z02 


p  is  a  point  on  the  original  circle 

P  is  a  point  on  the  new  circle  corresponding  to  p. 

Instead  of  adding  iiri  to  d,  add  i\  to  — ,  so  that  Oe  represents 

1  J.    SUMEC,    E.  T.  Z.,  Feb.  3,  1910.     The  method  is  due  to  MESSRS. 
STEHR  AND  PICHELMEYER. 
3 


34  INDUCTION  MOTOR 

both  ii  and  the  resistance  component  to  be  added  to—-     Then 

Fig.  21, 

A  OP  A  ~  A  OPc  ~  ArOA 
AP  :AO  ::AO  :  Ar 

.'.AP-Ar  =   ffil1  (21) 


As  •  Ar  =  AB2  =  z02  from  a  well-known 
property  of  the  circle. 

I  o       A 

(22) 


Zo2 


Hence,  P  lies  on  the  radius  vector  from  A  to  c  intersecting  the 
circle  in  s,  and  P  lies  again  on  a  circle  as  any  arbitrary  radius 
vector  AP  is  always  proportional  to  the  radius  vector  As. 

Now,  we  have 


(23) 


Now,  from  Fig.  22,  we  have 

Ag:AO::AO:Ak 
.'.    Ag-Ak  =  AO2 

But,  from  the  properties  of  the  circle, 
Ad-Ak=  zQ2 
.  Ag       AO2 
*'  Ad        z02 

However,  Agr  :  Ad  ::  AC0  :  AC 


(24) 


AC.-AC.«  (25) 

w 

From  inspection  of  Fig.  22,  calling  D  the  diameter  of  circle 
with  C  as  center,  and  D0  the  diameter  of  circle  with  C0  as  center, 
Do  :D  ::AC0  :  AC 


(26) 


THE  GENERAL  ALTERNATING-CURRENT  TRANSFORMER    35 


ce    g 


a  -e 


36 


INDUCTION  MOTOR 


To  obtain  the  coordinates  of  the  center  of  the  circle  of  diameter 
DO,  which  is  the  new  locus  of  the  primary  current  with  full 
consideration  of  the  primary  resistance  of  the  transformer,  we 
proceed  as  follows : 

Cob  :AO  ::(AC  -  ACQ)  :  AC 


C»6  :  g 


::  AC 


:AC 


(27) 


The  abscissa  Ob  =  OC  -  bC  is  found, 

bC  :OC  ::CCQ:AC 

bC  :OC  ::(AC  -  AC,}  :  AC 


.:  bC  =  OC 


Ob  =  £  = 


It  is  interesting  to  note  that  point  g  in  the  two  diagrams  of  Fig. 
22  remains  a  fixed  point  through  which  all  vectors  of  the  second- 
ary currents  may  be  drawn  from  the  corresponding  points  of  the 
vector  locus  of  the  primary  current  with  C0  as  center.  The 
proof  of  this  must  be  left  to  the  reader. 

This  method  has  been  used  to  consider  a  most  instructive  case, 
viz.,  that  of  a  transformer  with  leakage  and  resistance  in  primary 
and  secondary  operating  upon  a  circuit  whose  impressed  voltage 
varies  proportional  to  the  frequency.  At  a  frequency  60  cycles 
per  second,  the  voltage  is  six  times  that  at  10  cycles,  etc.  If 
there  were  no  primary  resistance,  the  locus  of  the  primary  cur- 
rent would  be  the  same  circle  through  the  entire  range  of  change 
of  frequency  and  voltage,  the  resultant  flux  remaining  the  same. 
If  there  is  primary  resistance,  however,  its  effect  will  be  greater 
with  reduced  primary  impressed  voltage  for  the  same  current. 


THE  GENERAL  ALTERNATING-CURRENT  TRANSFORMER   37 


The  characteristic  circles  for  this  case  are  drawn  in  Fig.  23  for 
the  conditions  given  in  the  following  table: 


FIG.  23. — Variable  frequency  transformer.     Primary  current  loci.     Impressed 
voltage  varies  with  frequency. 

VARIABLE  FREQUENCY  TRANSFORMER 


61 

TI 

er 

zo2 

Do 

r 

£ 

50 

2,500 

3,600 

69.50 

15.20 

41.7 

40 

1,600 

2,700 

59.30 

16.30 

35.6 

30 

900 

2,000 

45.00 

16.50 

27.0 

20 

400 

1,500 

26.65 

14.70 

16.0 

10 

100 

1,200 

8.35 

9.17 

5.0 

0 

0 

1,100 

0.00 

0.00 

0.0 

io  =  10                                                   D  =  100 

a   =  .1                                                    f!  =  1 

38 


INDUCTION  MOTOR 


These  results  will  be  found  of  great  help  in  understanding 
Chap.  XII  on  "  Concatenation  of  Induction  Motors."  It  is 
also  of  importance  where  induction  motors  are  started  with  their 
generators  from  standstill  by  gradually  raising  the  speed  with 
constant  field  excitation. 

We  shall  now  investigate  whether  it  is  allowable  to  neglect 
the  primary  resistance  in  practice  so  far  as  it  extends  to  its 
influence  upon  the  circle  locus  of  the  primary  current. 

We  shall  calculate  five  values  for  the  percentage  of  voltage 
consumed  by  resistance  and  tabulate  the  errors  in  the  coordinates 
of  the  centers  of  the  circles.  We  shall  assume  a  leakage  coeffi- 
cient a  =  0.06,  corresponding  to  a  maximum  obtainable  power 
factor  of  0.893.  We  will  assume  the  transformer  or  motor  to 
operate  at  this  point  of  maximum  power  factor,  and  therefore 
the  normal  current  will  be  50  amp.  for  i0  =  12  amp. 

TABLE  OF  ERRORS  INTRODUCED  BY  NEGLECTING  PRIMARY  RESISTANCE 


iiTi 
ei 
per  cent 

TI 

ei 
ri 

& 

z0* 

£>o 

f 

$ 

1 

2 

0.2 

2,500 

625  •  104 

625.  255-  104 

200.0 

0.00 

112.0 

3 

0.3 

1,670 

280  •  104 

280.  255  -104 

200.0 

0.00 

112.0 

5 

0.5 

1,000 

100  -104 

100.  255  -104 

199.5 

3.00 

111.6 

7 

0.7 

715 

51  •  104 

51.  250-10* 

199.0 

3.58 

111.3 

10 

1.0 

500 

25  •  104 

25.255-104 

198.0 

4.95 

109.0 

ff  =  0.06 
io  =  12  amp. 
D  =  200  amp. 


ii  =  50  amp. 
Vi  =0.2  ohm 
ev  =  500  volts 


From  this  table  it  appears  that  up  to  5  per  cent  the  effect  of 
resistance  upon  the  circle  is  negligible  without  a  question  of  a 
doubt,  while  from  5  per  cent  to  10  per  cent  it  seems  negligible 
for  all  practical  purposes,  errors  from  other  sources  being  vastly 
greater  than  from  the  neglect  of  primary  resistance,  so  far  as 
the  magnitude  and  location  of  the  circle  are  concerned. 

Needless  to  say,  the  losses  due  to  primary  resistance  must 
not  be  neglected.  How  this  is  done  is  shown  below. 

ACCOUNTING  FOR  PRIMARY  RESISTANCE  BY  THE 
METHOD  OF  RECIPROCAL  VECTORS 

In  Chap.  VIII  of  the  first  edition  of  this  book  the  circle  dia- 
gram of  the  alternating-current  transformer  was  developed,  in- 


THE  GENERAL  ALTERNATING-CURRENT  TRANSFORMER    39 

eluding  the  effect  of  primary  resistance,  using  the  well-known 
method  of  reciprocal  vectors  first  applied  to  this  problem  by  Prof. 
F.  Bedell  and  A.  C.  Crehore.1  Though  principally  of  academic 
interest,  we  repeat  here  the  demonstration  as  a  useful  exer- 
cise in  the  geometric  interpretation  of  alternating-current 
phenomena. 


Oa.Ob 
Od.Oe  =  z02 
Oa.Ob  =  Od.  Oe 
Oa  .  Od  :  :  Oe  :  Ob 
FIG.  24. — Reciprocal  vectors. 

Figure  24  from  similar  triangles  A  Odb  and  A  Oae  we  have 

Oa  :0e  ::0d  :0b 

.'.     Oa  Ob  =  OdOe  =  constant  (28) 

Also  Og Oh  =  constant 

(Oc  -  p)(0c  +  p)  =  Oc2  -  p2  =  *o2 

.'.     OaOb  =  OdOe  =  z02  (29) 

where  z0  is  the  length  of  the  tangent  to  the  circle  from  0  to  c. 

1  (1)  F.  BEDELL  and  A.  C.  CREHORE,  ''Resonance  in  Transformer  Cir- 
cuits," Physical  Review,  Vol.  ii,  No.  12,  May- June,  1895. 

(2)  FREDERICK  BEDELL,  "The  Principles  of  the  Transformer,"  p.  223 
el.  seq.     New  York,  The  Macmillan  Company,  1896. 


40 


INDUCTION  MOTOR 


Now,  imagine  a  circle  (Fig.  25)  about  C  as  center  representing 
the  locus  of  the  primary  impressed  e.m.f.  for  a  constant-current 
transformer  whose  current  vector  coincides  with  the  ordinate  OA 
as  proved  in  Chap.  II.  If,  instead  of  a  constant  current,  we 
keep  constant  the  impressed  e.m.f.,  then  the  current  of  the  trans- 
former will  maintain  the  same  phase  relation  to  its  impressed 

e.m.f.,  but  its  magnitude  will  be  increased  in  the  ratio  of  -j-,  or 
--  and  its  phase  will  remain  ^,  so  that  the  variable  voltages  for 


Oc  \0g  ::  Oc':  Od' 
Og'-  Og'=0d'*  Od=500 

FIG.  25. — Transformation  from  constant  current  to  constant  voltage  by  means 
of  reciprocal  vectors. 

the  constant-current  transformer  will  be  represented  by  vectors 
on  the  left  of  OA,  while  the  variable  currents  for  the  constant 
potential  transformer  will  be  represented  by  vectors  on  the  right 
of  OA. 
Now, 


A 
Oa 


Ob 


•/!  =  Oa' 

/!    =    Ob' 


£•'• '  Oc> 


£ '»  =  Og' 

Od' 


—•/i 
Od   l 


THE  GENERAL  ALTERNATING-CURRENT  TRANSFORMER  41 


where  I\  is  the  primary  current  of  the  constant-current  trans- 
former. 

Add  the  last  two  equations, 

-  Od) 


Od-Og 


=  Od'  +  Og' 


=  20C' 


(30) 


Od-Og 

This  equation  (30)  can  also  be  written  by  substituting  the  value 
1         Od' 


OC-Od'  =  OC'Og  (31) 

Either  equation  can  be  used  to  calculate  the  center  of  the  derived 
circle. 


C'P'      OC'      k* 

-CQ  =  oc  =7*  =  const- 

FIG.  26. — Reciprocal  vectors. 

We  will  now  prove,  for  the  sake  of  completeness,  that  the 
inverse  of  a  circle  is  another  circle.  Let  P,  Fig.  26,  be  any  point 
on  the  circle,  P'  its  inverse.  Let  OP  cut  the  circle  again  in  Q. 
Let  C  be  the  center  of  the  circle.  Then  OP  OP'  =  k2,  where 
k2  is  ei/i  in  the  preceding  argument.  Now  OP  OQ  =  z2,  where 
z  is  tangent  to  circle  C  from  0. 

OP'  _  k2 
''•  OQ   ~  ^ 

OC'      k2 
Take  C'  on  OC  such  that  QJT  =  ~2>  *nen 

'and  CPf  is  parallel  to  CQ. 
Therefore,  C'P'       OC' 


CQ 


fc2 
=  -=  =  constant 


42 


INDUCTION  MOTOR 


Therefore,  Pf  describes  a  circle  round  C"  and  we  have  proved 
that  the  inverse  of  a  circle  is  another  circle.  This  proof,  it  will 
be  noticed,  is  similar  to  that  given  previously. 

THE  LOSSES  AND  THEIR  REPRESENTATION  BY  STRAIGHT  LINES 

The  first  and  second  methods  showing  the  transformation 
of  the  resistance-less  circle  locus  into  a  circle  locus  taking  ac- 
count of  primary  resistance,  yield  a  better  insight  into  the 
transformation  than  does  the  third  method. 

In  order  to  render  an  account  of  the  loss  due  to  primary 
resistance  we  shall  first  consider  the  circle  diagram  with  the 


FIG.  27.  —  Accounting  for  secondary  copper  loss  by  means  of  the  loss  line. 

center  of  the  circle  on  the  abscissa  axis,  i.e.,  we  shall  assume,  as 
we  have  demonstrated  in  Chap.  Ill,  that  the  circle  locus  is 
changed  only  immaterially  by  the  presence  of  primary  resistance 
of  such  an  order  as  is  encountered  in  practical  apparatus. 

The  primary  copper  loss  is  equal  to  ifti,  the  secondary  copper 

loss  is  equal  to  e^-r^.     The  vector  i\  =  c,  i 

Fig.  27.     Obviously, 

a  :x  :  :D  :a 


b,  and  —  =  o, 


or 
or 
or 


a2  =  x-D 
*r  =  x(Dr) 


(32) 


THE  GENERAL  ALTERNATING-CURRENT  TRANSFORMER   43 

In  other  weirds,  -  — ,  which  is  the  watt-component  corresponding 

to  the  loss  in  the  secondary  copper,  may  be  represented  by  the 
ordinates  of  the  straight  line  dg. 

This  representation  is  evidently  applicable  only  so  long  as  a  =  —  > 

v\ 

or  a  proportional  to  — ,    which  relation  can  be  proved  to  hold 

even  for  the  case  in  which  primary  resistance  ri  is  taken  account 
of. 


FIG.  28. — Accounting  for  primary  copper  loss  by  means  of  the  loss  line. 


The  primary  copper  loss  is  found  as  follows,  Fig.  28, 

C2  =  a2  +  62  +  2ab  cosd 

c2  =  xD  +  62  -f-  2bx 

.'.  c2  =  x(D  +  26)  -f  62 

Also  t'iVi  =  z(D  +  2i0)ri  H 


Also 


(33) 
(34) 

(35) 


In  words,  the  primary  copper  loss  is  proportional  to  x  plus  the 
constant  *o2ri,  and  it  may  therefore  be  represented  by  a  straight 
line  whose  ordinates,  measured  from  the  z-axis  are  equal  to  the 
watt-component  of  the  primary  copper  loss. 


44  INDUCTION  MOTOR 

For  ii  =  i0  we  obtain 

x(D  +  2*0)7-1  =  0 
.*.z   =0 
For  ii  =  0  we  obtain 


These  simple  methods  for  determining  the  copper  losses  were 
originally  given  by  the  author  20  years  ago  in  Appendix  III  of  the 
first  edition  of  this  book. 

If  the  position  of  the  circle  is  such  that  the  center  does  not  lie 
on  the  abscissa,  then  the  watt-components  of  the  losses  are  no 
longer  to  be  measured  parallel  to  e\  but  as  proved  above,  normal 
to  the  diameter  of  the  circle. 

THE  IRON  LOSSES  DUE  TO  HYSTERESIS  AND  EDDY  CURRENTS 

The  iron  losses,  or  the  core  loss,  of  an  induction  motor  are  due 
to  hysteresis  and  eddy  currents.  There  has  been  discussion 
regarding  the  most  accurate  manner  in  which  to  take  them  into 
account  in  the  diagram. 

_  D 


FIG.  29. — Equivalent  circuits  show-  FIG.    30. — Equivalent   circuits   show- 

ing position  of  core  loss  circuit.  ing   position  of  core  loss  circuit.     (Al- 

ternate.,) 

In  view  of  the  fact  that  the  hysteresis  loss  is  likely  to  be  pro- 
portional to  the  resultant  primary  field,  this  loss  may  well  be 
assumed  constant.  The  loss  due  to  the  eddy  currents  generated 
by  this  field  may  also  be  assumed  constant  at  all  loads.  Losses 
due  to  stray  fields  are  apt  to  be  very  considerable  and  these, 
therefore,  would  increase  with  increasing  current  load.  As  it  is 
not  practicable  to  take  all  these  factors  into  account,  I  proposed 
first  in  1896,  and  I  was  seconded  by  Prof.  Blondel,  to  look  upon 
these  losses  as  though  produced  in  a  resistance  shunted  across 
the  primary  potential,  Fig.  29.  Other  writers,  like  Steinmetz, 
Arnold,  LaCour,  McAllister,  and  Bragstad  have,  however,  used 
a  second  method  of  an  equivalent  circuit,  as  shown  in  Fig.  30,  in 
which  at  standstill  the  core  loss  is  a  minimum,  gradually  in- 
creasing with  decreasing  load  or  increasing  speed.  This  theory 


THE  GENERAL  ALTERNATING-CURRENT  TRANSFORMER   45 

does  not  appear  very  reasonable,  as  the  hysteresis  loss  is  more 
likely  to  be  dependent  upon  the  total  field  rather  than  the  com- 
mon field.  It  is  true  that  it  is  much  more  difficult  to  take  into 
account  the  core  loss  if  it  depended  upon  the  potential  at  the 
terminals  of  the  exciting  shunt  as  it  is  indicated  in  the  equivalent 
circuit.  But  the  fact  that  it  is  more  difficult  to  take  it  into 
account  with  this  assumption,  though  this  assumption  is  farther 
removed  from  the  actual  conditions,  should  be  no  reason  why 
it  should  be  considered  necessary  to  do  it.  "  Error  which  is  not 
pleasant,  is  surely  the  worst  form  of  wrong." 


FIG.  31. 

A  third  method  which  I  have  followed  from  time  to  time  since 
1900  assumes  that  the  circle  diagram  is  derived  without  taking 
into  account  the  core  losses,  and  that  this  loss  is  later  accounted 
for  by  deducting  it  from  the  secondary  output.  This  procedure 
is  likely  to  be  about  as  accurate  as  any  of  the  previous  methods 
and  it  has  the  merit  of  greater  simplicity.  I  believe  it  was  first 
suggested  by  my  friend  Heyland. 

Load  losses  should  doubtless  be  accounted  for  by  an  increase 
in  the  primary  and  secondary  resistances.  If  this  is  done,  then 
it  appears  altogether  illogical  to  take  into  account  the  core  loss 
by  making  it  depend  upon  the  common  field,  which  goes  through 


46  INDUCTION  MOTOR 

the  air-gap,  and  this  strange  conception  would  never  have  arisen 
but  for  the  equivalent  circuit  methods  which  make  it  appear  as 
though  the  voltage  drop  in  the  primary  leakage  reactance  oc- 
curred outside  the  machine  while,  in  reality,  the  flux  which 
produces  the  reactance  voltage  ,  is  vectorially  added  to  the  com- 
mon air-gap  flux. 

If  we  assume  the  core  loss  which  is  made  up  of  hysteresis  and 
eddy-current  losses  to  be  constant  at  all  loads,  which  is  a  very 
problematical  assumption,  and  justifiable  only  on  account  of  our 
profound  ignorance  of  the  causes  of  core  loss  and  the  magnitude 
of  these  losses,  then  we  may  assume  its  effect  to  be  equivalent 
to  a  constant  watt  loss,  whose  watt-component  is  constant  and 
may  be  represented  in  our  diagram  by  a  line  parallel  to  the  dia- 
meter of  the  circle.  By  this  amount  the  available  secondary 
power  will  be  diminished.  We  will  recur  to  this  matter  in 
subsequent  chapters,  while  Fig.  31  shows  these  losses  graphically. 

B.  THE  TRANSFORMER  WITH  INDUCTIVE  LOAD 

Assume  the  secondary  of  a  transformer  to  be  closed  by  a 
circuit  with  resistance  and  inductance.  Assume  inductance 
and  resistance  to  vary  in  such  a  manner  that  the  power  of  the 
secondary  external  circuit  remains  constant.  Then,  Fig.  32, 
we  have: 

F2  the  secondary  resultant  magnetic  field  which  induces 


iz  the  secondary  load  current  lagging  in  time  by  the  angle  ty. 
/2  =  (vz  —  1)$2  =  ab,  secondary  leakage  field  in  phase  with 

the  secondary  load  current. 
$2  =  be  the  fictitious  secondary  flux  proportional  to  the  sec- 

ondary m.m.f  . 
=  ae 
<f>i  =  Oe    the    fictitious   primary   flux   proportional    to   the 

primary  m.m.f. 
/i  =  (vi  —  l)$i  =  be  the  primary  leakage  flux. 


dc 


and  Amed  are  similar  triangles.     Angle  Oae  is  equal  to  a 
right  angle  plus  i/%  it  is  therefore  a  constant  angle  for  a  variation 


THE  GENERAL  ALTERNATING-CURRENT  TRANSFORMER   47 


of  1%.     Hence  point  a  moves  on  the  arc  of  a  circle,  and  point  e 
does  equally  so. 

Od  :ad  : :  md  :  de 


—  :D  = 


Od 

S3 


_ 

"'1 


1N2      ~ 

-  1)— -D 

t>l 


(37) 
(38) 
(39) 


(T-.44 


Induction  Motor  Range  with 

^- — ^ 

uctance  in  Sec-Circul 


FIG.  32. — Inductance    in    the    secondary    of    constant    potential   transformer 
(motor   range)    and    capacity   in    secondary   for   induction   generator   range. 

The  effect  of  the  inductance  in  the  secondary  circuit  consists 
in  greatly  reducing  the  maximum  watt-component  of  the  primary 
current,  and  in  reducing  the  power  factor.  The  capacity  of  a 
transformer  is  thus  greatly  reduced  by  an  inductive  load. 

C.  THE  TRANSFORMER  WITH  CAPACITY  LOAD 

Assume  the  secondary  of  a  transformer  to  be  closed  by  a  cir- 
cuit with  such  condenser  capacity  that  both  resistance  and  ca- 
pacity may  be  varied  so  as  to  keep  the  power  factor  cos  ^2  of 
the  external  circuit  constant.  Then  Fig.  33  shows  that  triangle 


48 


INDUCTION  MOTOR 


Oad  is  similar  to  triangle  med  and  eb  =  $2,  in  phase  with  the 
secondary  current  i'2. 

Od:ad::D:de  (39) 


(40) 
(41) 
(42) 


Induction    Motor  Range  with 
Capacity  in  Secondary  Circuit 


Induction  Generator  Bange 
Inductance  in  Secondary  Circuit 


FIG.  33. — Capacity  in  the  secondary  of  constant  potential  transformer  (motor 
range)  and  inductance  in  secondary  for  induction  generator  range. 

The  effect  of  variable  capacity  in  the  secondary  of  a  constant- 
potential  transformer  consists  therefore  in  greatly  raising  its 
receptive  capacity  and  in  increasing  the  power  factor. 

How  this  effect  can  be  obtained  by  means  of  rotating  apparatus 
is  shown  in  Chap.  XV.  The  effect  of  such  apparatus  upon  the 
characteristics  of  the  induction  generator  is  discussed  in  Chap.  IX. 


CHAPTER  IV 
THE  MCALLISTER  TRANSFORMATIONS 

It  is  well  known  that  geometrical  figures  can  be  "  transformed" 
by  means  of  a  complex  function  used  as  an  operator  and  that  the 
"transformation"  is  frequently  a  solution  of  a  problem  otherwise 
impossible  of  solution.  There  are  certain  partial  differential 
equations  in  physics,  especially  in  hydrodynamics  and  in  elec- 
tricity, to  which  these  transformations  have  been  applied  success- 
fully. In  fact,  the  entire  fascinating  subject  of  "conformal 
representation"  of  functions  including  the  problem  of  "Merca- 
tor's  Projection"  in  geography  and  solutions  for  the  electrostatic 
capacity  of  conductors  of  different  shape,  is  intimately  connected 
with  this  subject 

In  connection  with  vector  diagrams  as  used  in  this  book,  the 
transformation  by  " reciprocal  vectors"  occurs  most  often  and 
it  has  therefore  been  discussed  in  the  previous  chapter.  It 
consists  in  simultaneously  shrinking  and  turning  a  vector,  and  it 
transforms  invariably  one  circle  into  another. 

As  a  general  rule  " point  for  point"  reductions  have  been 
made  both  in  the  graphical  treatment  and  in  the  analytical  treat- 
ment by  means  of  complex  algebra.  In  a  few  cases,  the  graphi- 
cal treatment  has  had  the  advantage  as  the  circle  locus  property 
permits  an  easy  representation  of  the  entire  set  of  complex  alge- 
braic equations.  Where,  however,  a  " point  for  point"  method 
has  to  be  resorted  to,  the  advantage  of  the  graphical  method 
is  less  apparent. 

It  has  been  shown  in  the  previous  chapter  that  the  addition 
of  resistance  in  series  with  the  induction  motor,  the  primary 
resistance  of  which  has  been  neglected,  leads  again  to  a  circle 
for  the  locus  of  the  primary  current.  The  center  of  this  circle 
is  raised  above  the  abscissa  and  its  diameter  is  smaller  than  that 
of  the  circle  representing  the  performance  of  the  motor  without 
primary  resistance.  However,  the  process  had  been  limited  to 
a  special  case  and  no  generalization  of  it  had  been  developed. 

Dr.  A.  S.  McAllister  has  succeeded  recently  in  applying  the 
same  process  of  reasoning  through  which  we  have  taken  the  reader 
4  49 


50 


INDUCTION  MOTOR 


in  the  development  of  the  general  circle  diagram,  to  the  general 
case  which  comprises  resistance  in  series,  reactance  in  series,  or 
resistance  and  reactance  in  series  with  the  motor.  He  has  dis- 
covered a  method,  simple  and  direct,  of  determining  the  "  center 
of  inversion  "  from  which  the  new  circle  can  be  located  by  drawing 
a  few  lines  to  the  image  of  the  original  circle.  It  may  be  re- 
marked in  passing  that  it  is  curious  that  so  simple  a  solution  has 
taken  25  years  to  be  brought  to  light. 

In  the  accompanying  figures  there  are  treated  the  following 
cases : 


A  and  QO 


Mirror  Plane 


FIG.  34. — The  McAllister  transformations.     Resistance  in  series  with  the  motor, 

without  core  loss. 

A.  Resistance  in  series  with  the  motor  without  core  loss  (Fig.  34). 

B.  Reactance  in  series  with  the  motor  without  core  loss  (Fig.  35)  . 

C.  Impedance  in  series  with  the  motor  without  core  loss    Figs.  36  and  37). 

D.  Resistance  in  series  with  the  motor  with  core  loss  (Fig.  38). 

E.  Reactance  in  series  with  the  motor  with  core  loss  (Fig.  39). 

F.  Impedance  in  series  with  the  motor  with  core  loss  (Fig.  40). 

It  is  interesting  to  observe  that,  if  we  assume  the  core  loss  to 
depend  upon  the  air-gap  field  F  only,  the  cases  D,  E,  and  .F  show 
that  the  resultant  primary  locus  remains  a  circle.  If  we  assume 
the  core  loss  to  depend  upon  the  total  primary  flux  FI,  which 


THE  MCALLISTER  TRANSFORMATIONS 


51 


seems  to  the  writer  the  more  rational  assumption,  then  the  re- 
sultant primary  locus  is  also  a  circle.  These  circles,  however, 
differ  in  magnitude  and  location. 

The  proof  of  the  McAllister  method  is  very  simple.  It  is 
suggested  in  all  the  diagrams.  The  start  is  made  with  the  origi- 
nal circle  whether  this  is  the  circle  of  the  locus  of  the  primary  cur- 
rent as  used  for  the  performance  of  the  induction  motor  in  this 
book,  or  whether  it  is  the  locus  of  the  primary  current  if  the 


FIG.  35. — The  McAllister  transformations.     Reactance  in  series  with  the  motor, 

without  core  loss. 

voltage  on  the  magnetizing  circuit  in  the  " equivalent"  diagram 
is  kept  constant. 

By  reflecting  this  circle  for  the  cases  A  and  D  below  the  ab- 
scissa; for  B  and  E  to  the  left  of  the  ordinate;  and  for  cases  C 
and  F  below  a  mirror  plane  which  forms  with  the  abscissa  an 
angle  whose  tangent  is  equal  to  X/R,  similar  triangles  are  formed 
by  drawing  rays  from  the  " center  of  inversion"  to  the  reflected 
point.  In  these  rays  lie  the  transformed  points  cutting  the  ray 
in  the  inverse  proportion. 


52 


INDUCTION  MOTOR 


The  procedure  then  is  the  same  in  all  six  cases. 

Referring  now  specifically  to  Fig.  34  illustrating  case  "A," 
the  circle  with  Cbr  as  radius  represents  the  locus  of  the  primary 
current  if  the  potential  on  the  •''magnetizing  circuit"  were  kept 

constant.  As  we  make  OA  equal  to  —  any  radius  vector  meas- 
uring the  primary  current  measures  also  the  ohmic  drop  in  the 
primary.  Thus,  as  we  have  seen  in  Figs.  22  and  26,  Chap.  Ill, 
the  point  A  or  00  becomes  the  center  of  inversion  from  which, 


00 


FIG.  36. — The  McAllister  transformations.     Impedance  in  series  with  the  motor, 

without  core  loss. 

by  means  of  similar  triangles  as  used  before,  we  obtain  the  new 
locus  for  the  primary  currents  with  center  C0  in  a  manner  indi- 
cated geometrically  in  the  figure.  The  vectors  drawn  from  0  to 
the  circle  whose  center  is  Co  represent,  in  the  scale  adopted, 
both  the  primary  current  and  the  primary  ohmic  drop. 

In  the  next  figure  (Fig.  35),  which  represents  the  case  "B" 
in  which  reactance  is  placed  in  series  with  the  motor,  the  process 
is  shown  in  greater  detail.  The  circle  with  C  as  center  repre- 
sents in  Op  the  primary  current  if  the  potential  on  the  magnet- 


THE  MCALLISTER  TRANSFORMATIONS 


53 


FIG.  37. — The  McAllister  transformations.     Impedance  in  series  with  the  motor, 

AC  =  Op. 


54 


INDUCTION  MOTOR 


izing  circuit  were  kept  constant.  As  OA  represents  the  primary 
impressed  voltage  divided  by  the  additional  reactance  placed 
in  series  with  the  motor,  Ac  =  Op  =  i\-Xi  represents  the  react- 
ance drop  in  the  circuit  connected  in  series  with  the  motor. 

Now  draw  the  circle  with  C"  as  center  which  is  theimage 
circle  of  C.  Note  that  AOpOO  =  AAcO  which  triangles  are  simi- 
lar to  ACO  -  00.  From  this  follows  that 


or 


Op  :  0  -  00  :  :  Ob  :  00  -  b 


00-b-Op  =  ~-i 

A  i 


A  and  00 


FIG.  38. — The  McAllister  transformations.     Resistance  in  series  with  the  motor, 

with  core  loss. 


This  proves  our  proposition  as  it  shows  that  the  point  b  lies  in- 
variably upon  the  ray  drawn  from  the  "  center  of  inversion 
00  to  the  "reflected"  point  p  or  b'  which  is  the  image  of  the 
original  point  p  in  the  mirror  plane  OA.  Point  b  divides  00  —  p 
in  such  a  ratio  that  the  product  of  the  vectors  drawn  from  00  is 
a  constant.  Hence  the  new  locus  of  ii  is  a  circle  with  Co  as 
center  CQb  being  perpendicular  to  00— p.  The  reader  would 


THE  MCALLISTER  TRANSFORMATIONS 


55 


FIG.  39. — The  McAllister  transformations.     Reactance  in  series  with  the  motor, 

with  core  loss. 


FIG.  40. — The    McAllister    transformations.     Impedence    in    series    with    the 
motor  with  core  loss. 


56  INDUCTION  MOTOR 

do  well  to  draw  for  himself  a  few  points  and  to  transform  them 
into  their  new  positions  in  the  vector  diagram. 

The  remaining  figures  referring  to  the  six  cases  enumerated  are 
sufficiently  clear  to  require  no  further  explanations  as  to  their 
mode  of  derivation,  provided  the  reader  will  take  the  pains  to 
draw  a  few  points  in  order  to  comprehend  the  principles  on  which 
this  method  is  based.  This  has  been  done  in  detail  in  Fig.  37 
illustrating  with  Fig.  36,  the  case  C. 

Similar  results  to  those  arrived  at  above  by  the  McAllister 
method  have  been  obtained  by  successive  applications  of  the 
method  of  reciprocal  vectors,  discussed  in  Chapter  III-A,  as 
shown  by  Messrs.  LaCour  and  Bragstad,  who  have  transformed 
the  admittance  circle  diagram  into  an  impedance  circle  diagram 
to  which  they  have  added  the  primary  impedance,  re-transform- 
ing the  new  impedance  diagram  back  into  the  final  admittance 
diagram.  This  method  is  described  at  length  in  the  great  work 
of  these  authors  frequently  referred  to  in  this  book. 


CHAPTER  V 
THE  ROTATING  FIELD  AND  THE  INDUCTION  MOTOR 

A.  THE  AMPERE  TURNS  AND  THE  FIELD  BELT 

In  the  induction  motor,  as  invented  by  Mr.  Nikola  Tesla,  two, 
three,  or  more  windings  lodged  in  slots,  usually  located  in  the 
stationary  part  or  stator,  are  fed  with  alternating  currents  of 
the  same  frequency  and  voltage  but  of  different  time-phase.  A 


FIG.  41. — Distributed  three-phase  winding.     The  belt  of  ampere-turns  and  flux. 

belt  of  these  windings  produces  a  rotating  field.  In  a  short-cir- 
cuited rotor  winding  currents  are  induced  whose  interaction 
with  the  field  results  in  the  production  of  torque.  We  shall 
briefly  examine  the  manner  in  which  such  a  field  is  produced. 

57 


58  INDUCTION  MOTOR 

Let  I,  II,  and  777  be  the  three  phases  of  the  stator.  We  shall 
assume  that  the  reluctance  of  the  iron  is  negligible  and  that  the 
induction  is  proportional  to  the  line  integral  of  the  m.m.fs., 
and  inversely  proportional  to  the  length  of  the  gap  (Fig.  41). 
We  shall  assume  the  current  to  vary  according  to  a  simple  sine 
law.  Then  the  intensity  of  the  currents  in  7  and  77  is  each  one- 
half  that  of  777.  The  m.m.fs.  of  each  phase  are  represented  by 
the  ordinates  of  the  curves  7,  77,  and  777  respectively.  Each 
ordinate  measures  the  m.m.f.  produced  in  that  point  of  the  cir- 
cumference where  it  is  drawn.  The  adding  together  of  the 
ordinates  of  the  three  curves  yields  the  heavy  line  curve  which  is 
the  sum  of  the  m.m.fs.  at  the  particular  moment  of  time  over  the 
circumference  of  the  air-gap. 

If  the  magnetic  reluctance  is  the  same  at  every  point  of  the 
circumference,  and  the  reluctance  of  the  iron  is  negligible,  then 
the  magnetic  induction  B,  produced  by  the  m.m.f.  belt  shown  in 
the  heavy  line  in  the  Fig.  41,  is  proportional  to  this  m.m.f.  and, 
therefore,  also  represented  by  the  ordinates  of  the  heavy  line 
curve.  We  call  the  total  flux  F,  and  we  assume  that  the  time 
variation  of  this  flux  follows  a  simple  sine  law.  We  shall  now 
calculate  the  e.m.f.  induced  by  this  flux  belt  in  each  of  the 
three  phases. 

We  shall  assume  arbitrarily  that  the  distribution  of  conductors 
per  phase  is  uniform,  in  other  words,  that  there  is  an  infinite 
number  of  slots. 

B.  THE  E.M.FS.  INDUCED  IN  THE  WINDINGS 

If  all  the  conductors  of  each  phase  were  concentrated  in  one 
slot,  then  the  e.m.f.  induced — remembering  that  two  conductors 
equal  one  complete  turn — would  be  according  to  Eq.  (10)  equal 
to  2.22  ^  -z-F-W~8  volts,  where  z  is  the  total  number  of  effective 
conductors  per  phase  equal  to  2n.  On  account  of  the  distribu- 
tion of  the  windings  over  one-third  of  the  pole-pitch,  only  the 
parts  of  the  flux  not  covered  with  hatchings  can  induce  an  e.m.f. 
according  to  this  formula,  while  the  hatched  parts  of  the  field 
will  have  a  considerably  smaller  effect.  Let  the  width  of  the 
coil  be  26,  and  n  conductors  in  the  coil  spread  over  26.  Per  unit 

Yl 

length  there  are,  therefore,  ^  conductors,  hence  the  element 
dx  contains  dx-    r  conductors.     The  number  of  lines  of  induction 


THE  ROTATING  FIELD  AND  THE  INDUCTION  MOTOR     59 

threading  all  the  conductors  in  the  element  dx  is  equal  to  Fx 
represented  by  the  hatched  area.     Hence, 

de  =  2.22-  ^•—•dx-Fx'lG-8  volts 

P  D      ^  D        " 

**x   =   &'n   —   &x'-^    = 


2  !    2  2        26 

=2-22— jp^-^i f^110-8 


e  =  2.22-~-^ 


26  I J0      26  Jo       26 

e  =  2.22-~- 


With  F  = 


e  =  2.22 
B-b 


2 
e  =  2.22  •~-^F)  10-s  (43) 


In  words,  the  e.m.f.  induced  by  the  field  F  upon  a  coil  of  the 
width  6  is  two-thirds  as  great  as  the  e.m.f.  which  would  be  induced 
by  the  same  field  upon  a  coil  whose  conductors  are  not  distributed 
but  lodged  in  one  slot.  Such  a  coil  would  not  produce  a  triangu- 
lar field  but  a  rectangular  field.  Therefore,  the  inductance  of 
the  flat  coil,  i.e.,  the  number  of  lines  or  tubes  of  force  per  unit 
current,  is  one-third  as  large  as  the  inductance  of  a  coil  lodged 
in  one  slot. 

The  e.m.f.  generated  by  the  field  belt  in  Fig.  41  can  now 
readily  be  calculated.     The  flux  of  the  white  area  is 


4 

The  e.m.f.  induced  by  this  flux  is  equal  to 
ea  =  2.22-~-z-  (^'^'t'b- 


The  hatched  areas  represent  a  flux  equal  to 
1  B 


The  e.m.f.  induced  by  this  flux  is 
eb  =  2.22-~.z--* 


60  INDUCTION  MOTOR 

Hence  e  =  ea  +  eb  =  2.22  ---  z(jjj.-  t-b-  B\lQr* 

wr% 

The  total  flux  is 


/.  e  =     .-~-z-- 

ZL 

e  ?=  2.  12-  —  2-F-10-8  (44) 

The  ampere-turns  in  each  phase  which  are  needed  to  produce 
the  induction  B  in  the  air-gap  are  determined  by  the  considera- 
tion, which  follows  immediately  from  Fig.  41,  that 

2(.4-wtM\/2)  =  #  2A 

where  A  is  the  single  distance,  or  length,  between  rotor  and  stator 
separated  by  the  air-gap,  n  the  number  of  conductors  per  pole 
per  phase,  and  iM  the  magnetizing  current.     The  reluctance  of  the 
iron  has  been  neglected. 
Hence, 

B  -  1  .  6A 


If  the  reluctance  of  the  iron  is  not  negligible,  then  the  magnetic 
induction  B  has  to  be  determined  point  for  point,  which  can  be 
done  with  the  aid  of  a  magnetizing  curve.  It  is  of  importance  to 
note  that  the  maximum  induction  does  not  extend  over  a  very 
large  part  of  the  pole-pitch,  hence  very  high  induction  in  the 
teeth  may  be  resorted  to  without  materially  raising  the  magnetiz- 
ing current,  although  increasing  the  losses  which  are  dependent 
upon  Bmax  in  the  teeth. 

There  are  numerous  modes  of  distribution  of  the  conductors 
per  pole  per  phase.  We  have  considered  above  that  each  phase 
covers  one-third  of  the  pole-pitch  in  a  three-phase  motor.  The 
conductors  may,  however,  spread  over  two-thirds  of  the  pole- 
pitch.  Figure  42  shows  the  m.m.f  .  belt  at  the  time  when  phase 
III  is  a  maximum. 

In  order  to  obtain  quickly  an  expression  of  the  e.m.f.  which 
such  a  field  induces,  we  avail  ourselves  of  the  simple  and  direct 
method  given  by  the  author  in  Appendix  II  of  the  first  edition  of 
this  book. 

Consider  a  winding  ag,  Fig.  43,  spanning  an  arc  of  180  electrical 
degrees,  i.e.,  extending  over  the  pole-pitch.  In  each  small  ele- 


THE  ROTATING  FIELD  AND  THE  INDUCTION  MOTOR    61 

ment  ab,  or  6c,  there  is  induced  an  e.m.f.  de,  represented  graphi- 
cally by  the   small  vector  AB   or  BC,   which  we  arbitrarily 


FIG.  42. — Field  belt  of  three-phase  motor.     Each  phase  spread  over  two-thirds 

of  pole  pitch. 


-A  •  Z    Effective  Conductors 
FIG.  43. — Winding  covering  pole  pitch. 

represent  at  right  angles  to  the  element.     Then  ABCD — G  is, 
so  to  speak,  the  hodograph  of  the  induced  voltages,  whose  vector 


62 


INDUCTION  MOTOR 


sum  is  AG.     If  x  is  the  total  number  of  conductors  in  the  winding 
ag,  then  there  are,  as  a  result  of  distribution, 

2 

-  •  z  effective  conductors  only. 

Hence,  e  =  \/2  '  ~  '  z  •  F  -  10~8  volts  (46) 

If  the  winding  is  distributed  over  two-thirds  the  pole-pitch, 
or  over  120  electrical  degrees,  Fig.  44,  then  the  number  of 
effective  conductors  is 

(Vz  -  H*)  z 

which  is  equal  to  0.825  z 

e  =  1.836  ~z-P'  10~8  volts  (47) 


FIG.  44. — Winding  covering  two-thirds  of  pole  pitch. 

In  a  quarter-phase  system  one-half  the  pole-pitch  is  covered 
by  the  coil,  Fig.  45,  therefore  there  are 

(\/2  -5-  2)  *  effective  conductors  per  phase 

.-.       e  =  2~-z  -F-  10-8  volts  (48) 

In  a  three-phase  motor,  the  coils  usually  cover  one-third  of 


THE  ROTATING  FIELD  AND  THE  INDUCTION  MOTOR    63 


Z    Effective  Conductors 


FIG.  45. — Winding  covering  one-half  of  pole  pitch. 


ffectfve  Conductors 


FIG.  46. — Winding  covering  one-third  of  pole  pitch. 


64  INDUCTION  MOTOR 

the  pole-pitch,  or  60  electrical  degrees  (Fig.  46).     Therefore, 
there  are 

(1  -r-  -^\  z  effective  conductors  per  pole  per  phase. 

.'.       e  =  2.12  '  ~-z-F  •  10-8  volts  (49) 

Ihese  methods  are  correct  whatever  the  shape  of  the  field 
belt,  as  long  as  F  equals  the  total  flux  produced. 

C.  THE  ELEMENTARY  THEORY  OF  THE  INDUCTION  MOTOR 

The  elementary  phenomena  of  the  functioning  of  an  induction 
motor  has  been  stated  with  admirable  clearness  by  Prof.  Gisbert 
Kapp,  to  whom  we  owe  so  much  in  the  interpretation  of  the 
theory  of  electrical  apparatus.  As  his  account  is  also  of  historical 
interest  it  is  here  reprinted  in  full,  as  was  done  in  Appendix  I  of 
the  first  edition  of  this  book. 

Excerpt  from  Gisbert  Kapp:  "Electric  Transmission  of  Energy, 
and  its  Transformation,  Subdivision,  and  Distribution. "  Fourth 
Edition,  Thoroughly  .Revised.  London,  Whittaker  &  Co., 
Paternoster  Square,  1894,  p.  301  to  p.  311. 

"In  order  to  be  able  to  deal  by  means  of  simple  mathematics  with 
the  working  condition  of  a  rotary  field  motor,  we  assume  that  the  induc- 
tion within  the  interpolar  space  between  field  and  armature  varies 
according  to  a  simple  sine  law.  Whether  this  induction  is  due  to  the 
current  in  the  field  coils  alone,  or  to  the  combined  effect  of  field  and 
armature  currents,  we  need  at  present  not  stop  to  inquire;  all  we  care 
to  know  is  that  such  an  induction  does  actually  exist  when  the  motor  is 
at  work,  and  that  the  sinusoidal  field  which  it  represents  revolves  with 
a  speed  corresponding  to  the  frequency  of  the  supply  currents.  Thus, 
if  there  be  four  field  coils,  and  the  frequency  is  50,  we  would  have  a  two- 
pole  field  revolving  50  times  a  second,  or  3,000  times  a  minute,  round 
the  centre  of  the  armature,  and  if  there  were  no  resistance  to  the  move- 
ment of  the  latter  it  would  be  dragged  round  by  the  field  at  a  speed  of 
3,000  r.p.m.  It  is  obvious  that  the  actual  speed  must  be  smaller.  If 
the  speed  of  the  armature  coincided  exactly  with  that  of  the  field,  then 
the  total  induction  passing  through  any  armature  coil,  or  between  any 
pair  of  conductors  on  the  armature  would  remain  absolutely  constant, 
and  there  would  be  no  e.m.f.,  and,  therefore,  no  current  induced  in  the 
armature  wires.  Where  there  is  no  current  there  can  be  no  mechanical 
force,  and  the  armature  could,  therefore,  not  be  kept  in  rotation.  In 
order  that  there  may  be  a  mechanical  force  exerted,  it  is  obviously 
essential  that  there  shall  be  a  variation  in  the  magnetic  flux  passing 


(Facing  page  64) 


v     ?•••••„    ' 


THE  ROTATING  FIELD  AND  THE  INDUCTION  MOTOR    65 


through  any  armature  coil,  and  that  necessitates  a  difference  in  the 
speed  of  rotation  between  field  and  armature.  This  difference  is  called 
the  'magnetic  slip'  of  the  armature.  If,  for  instance,  the  speed  of  the 
field  in  our  two-pole  motor  is  50  revolutions  per  second,  and  the  speed 
of  the  armature  48  revolutions  per  second,  we  would  have  a  magnetic 
slip  of  two  revolutions  out  of  50,  or  4  per  cent.  In  modern  machines  the 
slip  at  full  load  averages  about  4  per  cent,  and  rarely  reaches  as  high 
as  10  per  cent,  so  that  good  rotary  field  motors  are  in  point  of  constancy 
of  speed  under  varying  load  about  equal  to  continuous  shunt  motors. 

"It  was  mentioned  above  that  the  motor  would  have  a  frequency  of 
50  revolutions  at  a  speed  only  by  4  per  cent  short  of  3,000  r.p.m.  This 
is  an  inconveniently  high  speed  for  any  but  very  small  sizes.  To  reduce 
the  speed  is,  however,  quite  easy.  We  need  only  increase  the  number, 
and  proportionately  reduce  the  length  of  the  field  coils.  Thus,  if 
instead  of  4  coils,  each  spanning  90°  of  the  circumference,  we  use  8  coils, 
each  spanning  45°,  and  connect  them  so  as  to  produce  two  rotary  fields, 
the  speed  will  be  reduced  to  one-half  of  its  former  value.  By  using 
12  coils  we  obtain  a  six-pole  motor,  in  which  the  speed  will  be  reduced 
to  one-third,  or  about  1,000  r.p.m.;  with  16  coils  we  get  down  to  750 
revolutions,  and  so  on.  In  order 
to  avoid  unnecessary  complexity 
we  shall,  however,  commence 
the  investigation  on  a  two-pole 
machine,  having  only  one  re- 
volving field,  and  leaving  the 
transition  to  a  multi-polar  ma- 
chine running  at  lower  speed 
until  the  more  simple  case  has 
been  dealt  with. 

"Such  a  machine  is  shown  in 
Fig.  47.  The  field  consists  of  a 
stationary  cylinder,  composed  of 
insulated  iron  plates,  and  pro- 
vided close  to  the  inner  cir- 
cumference with  holes  through 
which  the  winding  passes.  The 

armature  is  also  a  cylinder  made  up  of  insulated  iron  plates  pro- 
vided with  holes  near  its  outer  circumference  for  the  reception  of  the 
conductors.  The  use  of  buried  conductors,  although  not  absolutely 
necessary,  has  two  important  advantages — first,  mechanical  strength 
and  protection  to  the  winding;  and,  secondly,  reduction  of  the  magnetic 
resistance  of  the  air-gap,  which,  it  will  be  seen  later  on,  is  an  essential 
condition  for  a  machine  in  which  the  difference  between  the  true  watts 
and  apparent  watts  shall  not  be  too  great.  The  armature  conductors 
may  be  connected  so  as  to  form  single  loops,  each  passing  across  a  diam- 


FIG.  47. — Diagram  of  rotor  and  stator  of 
induction  motor. 


66  INDUCTION  MOTOR 

eter,  or  they  may  all  be  connected  in  parallel  at  each  end  face  by  means 
of  circular  conductors,  somewhat  in  the  fashion  of  a  squirrel  cage. 
Either  system  of  winding  does  equally  well,  but  as  the  latter  is  mechan- 
ically more  simple,  we  will  assume  it  to  be  adopted  in  Fig.  47.  The 
circular  end  connections  are  supposed  to  be  of  very  large  area  as  com- 
pared with  the  bars,  so  that  their  resistance  may  be  neglected.  The 
potential  of  either  connecting  ring  will  then  remain  permanently  at 
zero,  and  the  current  passing  through  each  bar  from  end  to  end  will  be 
that  due  to  the  e.m.f.  acting  in  the  bar  divided  by  its  resistance.  It  is 
important  to  note  that  the  e.m.f.  here  meant  is  not  only  that  due  to 
the  bar  cutting  through  the  lines  of  the  revolving  field,  but  that  which 
results  when  armature  reaction  and  self-induction  are  duly  taken  into 
account. 

"Let  us  now  suppose  that  the  motor  is  at  work.  The  primary  field 
produced  by  the  supply  currents  makes  ~i  complete  revolutions  per 
second,  whilst  the  armature  follows  with  a  speed  of  ~2  complete  revolu- 
tions per  second.  The  magnetic  slip  is  then 

/***/  I        —        /"X^/O 

s  =  -  (50) 


If  the  field  revolves  clockwise,  the  armature  must  also  revolve  clock- 
wise, but  at  a  slightly  slower  rate.  Relatively  to  the  field,  then,  the 
armature  will  appear  to  revolve  in  a  counter  clockwise  direction,  with 
a  speed  of 

~  =  ~i  —  ~2  (51) 

revolutions  per  second.  As  far  as  the  electro-magnetic  action  within 
the  armature  is  concerned,  we  may  therefore  assume  that  the  primary 
field  is  stationary  in  space,  and  that  the  armature  is  revolved  by  a  belt 
in  a  backward  direction  at  the  rate  of  ~  revolutions  per  second.  The 
effective  tangential  pull  transmitted  by  the  belt  to  the  armature  will 
then  be  exactly  equal  to  the  tangential  force  which  in  reality  is  trans- 
mitted by  the  armature  to  the  belt  at  its  proper  working  speed,  and  we 
may  thus  calculate  the  torque  exerted  by  the  motor  as  if  the  latter  were 
worked  as  a  generator  backward  at  a  much  slower  speed,  the  whole  of  the 
power  supplied  being  used  up  in  heating  the  armature  bars.  The  object 
of  approaching  the  problem  from  this  point  of  view  is  of  course  to  simplify 
as  much  as  possible  the  whole  investigation.  If  we  once  know  what 
torque  is  required  to  work  the  machine  slowly  backward  as  a  generator, 
it  will  be  an  easy  matter  to  find  what  power  it  gives  out  when  working 
forward  as  a  motor  at  its  proper  speed. 

"Let,  in  Fig.  48,  the  horizontal  a,  c,  b,  d,  a,  represent  the  interpolar 
space  straightened  out,  and  the  ordinates  of  the  sinusoidal  line,  B,  the 
induction  in  this  space,  through  which  the  armature  bars  pass  with  a 
speed  of  ~  revolutions  per  second.  We  make  at  present  no  assumption 


THE  ROTATING  FIELD  AND  THE  INDUCTION  MOTOR    67 


as  to  how  this  induction  is  produced,  except  that  it  is  the  resultant  of 
all  the  currents  circulating  in  the  machine.  We  assume,  however,  for 
the  present  that  no  magnetic  flux  takes  place  within  the  narrow  space 
between  armature  and  field  wires,  or,  in  other  words,  that  there  is  no 
magnetic  leakage,  and  that  all  the  lines  of  force  of  the  stationary  field 
are  radial.  The  rotation  being  counter  clockwise,  each  bar  travels  in 
the  direction  from  a  to  c  to  b,  and  so  on.  The  lines  of  the  field  are 
directed  radially  outwards  in  the  space  dac,  and  radially  inward  in  the 
space  cdb.  The  e.m.f.  will,  therefore,  be  directed  downwards  in  all  the 
bars  on  the  left,  and  upwards  in  all  the  bars  on  the  right  of  the  vertical 
diameter  in  Fig.  47.  Let  E  represent  the 
curve  of  e.m.f.  in  Fig.  48,  then,  since  there 
is  no  magnetic  leakage,  the  current  curve 
will  coincide  in  phase  with  the  e.m.f. 
curve,  and  we  may  represent  it  by  the 
line  7.  It  is  important  to  note  that  this 
curve  really  represents  two  things.  In 
the  first  place,  it  represents  the  instanta- 
neous value  of  the  current  in  any  one  bar 
during  its  advance  from  left  to  right;  and 
in  the  second  place  it  represents  the  per- 
manent effect  of  the  current  in  all  the 
bars,  provided,  however,  the  bars  are 
numerous  enough  to  permit  the  repre- 
sentation by  a  curve  instead  of  a  line 
composed  of  small  vertical  and  horizontal 
steps.  The  question  we  have  now  to  in- 
vestigate is:  What  is  the  magnetising 
effect  of  the  currents  which  are  collectively  represented  by  the 
curve  7?  In  other  words,  if  there  were  no  other  currents  flowing 
but  those  represented  by  the  cur.ve  7,  what  would  be  the  disposition 
of  the  magnetic  field  produced  by  them?  Positive  ordinates  of  7 
represent  currents  flowing  upwards  or  towards  the  observer  in  Fig. 
47,  negative  ordinates  represent  downward  currents.  The  former  tend 
to  produce  a  magnetic  whirl  in  a  counter  clockwise  direction,  and  the 
latter  in  a  clockwise  direction.  Thus  the  current  in  the  bar  which 
happens  at  the  moment  to  occupy  the  position  b,  will  tend  to  produce 
a  field,  the  lines  of  which  flow  radially  inwards  on  the  right  of  6,  and 
radially  outwards  on  the  left  of  b.  Similarly  the  current  in  the  bar 
occupying  the  position  a  tends  to  produce  an  inward  field,  i.e.,  a  field 
the  ordinates  of  which  are  positive,  in  Fig.  48,  to  the  left  of  a,  and  an 
outward  field  to  the  right  of  a.  It  is  easy  to  show  that  the  collective 
action  of  all  the  currents  represented  by  the  curve  7  will  be  to  produce  a 
field  as  shown  by  the  sinusoidal  line  A.  This  curve  must  obviously 
pass  through  the  point  6,  because  the  magnetizing  effects  on  both  sides 


FIG.  48.  —  The  interpolar 
space  of  the  induction  motor. 
The  m.m.f.  and  flux  belts. 


68  INDUCTION  MOTOR 

of  this  point  are  equal  and  opposite.  For  the  same  reason  the  curve 
must  pass  through  a.  That  the  curve  must  be  sinusoidal  is  easily 
proved,  as  follows :  Let  i  be  the  current  per  centimeter  of  circumference 
in  fe,  and  let  r  be  the  radius  of  the  armature;  then  the  current  through 
a  conductor  distant  from  b  by  the  angle  a,  will  be  i  cos  a  per  centimeter 
of  circumference.  If  we  take  an  infinitesimal  part  of  the  conductor 
comprised  within  the  angle  da,  the  current  will  therefore  be 

di  =  ir  cos  a  da 

and  the  magnetizing  effect  in  ampere-turns  of  all  the  currents  comprised 
between  the  conductor  at  b,  and  the  conductor  at  the  point  given  by  the 
angle  a  will  be 

di  =  —ir  sin  a  (52) 


J 


and  since  the  conductors  on  the  other  side  of  b  act  in  the  same  sense, 
the  field  in  the  point  under  consideration  will  be  produced  by  2ir  sin  a 
ampere-turns,  i  being  the  current  per  centimeter  of  circumference  at  b. 

"Since  for  low  inductions,  which  alone  need  here  be  considered,  the 
permeability  of  the  iron  may  be  taken  as  constant,  it  follows  that  the 
field  strength  is  proportional  to  ampere-turns  and  that  consequently  A 
must  be  a  true  sine  curve. 

"When  starting  this  investigation,  we  have  assumed  that  the  field 
represented  by  the  curve  B  is  the  only  field  which  has  a  physical  exist- 
ence in  the  motor;  but  now  we  find  that  the  armature  currents  induced 
by  B  would,  if  acting  alone,  produce  a  second  field,  represented  by  the 
curve  A.  Such  a  field,  if  it  had  a  physical  existence,  would,  however, 
be  a  contradiction  of  the  premise  with  which  we  started,  and  we  see 
thus  that  there  must  be  another  influence  at  work  which  prevents  the 
formation  of  the  field  A.  This  influence  is  exerted  by  the  currents  pass- 
ing through  the  coils  of  the  field  magnets.  The  primary  field  must 
therefore  be  of  such  shape  and  strength,  that  it  may  be  considered  as 
composed  of  two  components,  one  exactly  equal  and  opposite  to  A, 
and  the  other  equal  to  B.  In  other  words,  B  must  be  the  resultant  of 
the  primary  field  and  the  armature  field  A.  The  curve  C  in  Fig.  48 
gives  the  induction  in  this  primary  field,  or  as  it  is  also  called,  the  "  im- 
pressed field,"  being  that  field  which  is  impressed  on  the  machine  by 
the  supply  currents  circulating  through  the  field  coils.  It  will  be 
noticed  that  the  resultant  field  lags  behind  the  impressed  field  by  an 
angle  which  is  less  than  a  quarter  period. 

"The  working  condition  of  the  motor,  which  has  here  been  investigated 
by  means  of  curves,  can  also  be  shown  by  a  clock  diagram.  Let  in 
Fig.  49,  the  maximum  field  strength  within  the  interpolar  space  (i.e., 
number  of  lines  per  square  centimeter  at  a  and  6  of  Fig.  47),  be  repre- 
sented by  the  line  OB,  and  let  01 a  represent  the  total  ampere-turns  due 
to  armature  currents  in  the  bars  to  the  left  or  the  right  of  the  vertical, 


THE  ROTATING  FIELD  AND  THE  INDUCTION  MOTOR    69 


then  OA  represents  to  the  same  scale  as  OB  the  maximum  induction 
due  to  these  ampere-turns.  We  need  not  stop  here  to  inquire  into  the 
exact  relation  between  01 a  and  OA,  this  will  be  explained 'later  on. 
For  the  present  it  is  only  necessary  to  note  that  under  our  assumption 
that  there  is  no  magnetic  leakage  in  the  machine,  OA  must  stand  at 
right  angles  to  OIa,  and  therefore 
also  to  OB,  and  that  the  ratio 
between  01 a  and  OA  (i.e.,  arma- 
ture ampere-turns  and  armature 
field)  is  a  constant.  By  drawing 
a  vertical  from  the  end  of  B  and 
making  it  equal  to  OA,  we  find 
OC  the  maximum  induction  of 
the  impressed  field.  The  total 
ampere-turns  required  on  the  field 
magnet  to  produce  this  im- 
pressed field  are  found  by  draw- 
ing a  line  from  C  under  the  same 
angle  to  CO,  as  AIa  forms  with 
AO,  and  prolonging  this  line  to 
its  intersection  with  a  line  drawn 
through  0  at  right  angles  to  OC. 
Thus  we  obtain  OIe,  the  total 
ampere-turns  to  be  applied  to 
the  field.  The  little  diagram 
below  shows  a  section  through 
the  machine,  but  instead  of  repre- 
senting the  conductors  by  little 
circles  as  before,  the  armature 
and  field  currents  are  shown 
by  the  tapering  lines,  the  thick- 
ness of  the  lines  being  supposed 
to  indicate  the  density  of  current  per  centimeter  of  circumference  at 
each  place." 

D.  THE  SQUIRREL  CAGE 

A  rotor  winding  consisting  of  a  number  of  bars  connected  in 
parallel,  or  short-circuited,  by  circular  end  rings,  was  invented 
by  the  late  M.  von  Dolivo-Dobrowolsky.  It  is  called  a  "  squirrel- 
cage"1  winding.  It  is  a  true  poly-phase  winding  and  its  theory 
and  understanding  are  based  on  the  same  principles  as  those  of 
other  poly-phase  windings. 

French,  "cage  d'ecureuil;"  German,  "Kurzschlussanker,"  or  "Kafig 
Wicklung." 


FIG.  49. — The  vector  diagram  of  the 
induction  motor  in  the  elementary 
theory. 

NOTE. — This  is  the  only  figure  in  this  book 
in  which  clockwise  rotation  has  been  assumed. 
It  was  taken  from  Mr.  Kapp's  book  published 
in  1894,  before  the  international  agreement 
had  been  reached  on  positive  counter-clock- 
wise rotation. 


70 


INDUCTION  MOTOR 


To  fix  ideas,  let  us  look  at  a  star-connected  generator  closed 
through  a  delta  connected  load,  Fig.  50.  In  each  phase  a  voltage 
e  is  induced.  The  delta  voltages  are  I-II,  II-III,  and  III-I. 
Calling  the  delta  voltages  E,  we  have 

E  =  \/3-e  (53) 


At  any  point  I,  II,  or  III,  the  algebraic  sum  of  the  currents  which 
meet  is  zero,  according  to  Kirchhoff  s  First  Law.  This,  ex- 
tended to  our  case,  may  be  expressed  in  vector  terms  and  we 


ii  + 


in 


Non-inductive  Voltage  Drop  in 
Balanced  Three-phase  System 

FIG.  50. 


Three-phase 
System 


FIG.  51. 


substitute  "vector  sum"  for  "algebraic  sum."     In  Fig.  51  this 
vector  sum  has  been  drawn,  from  which  we  see  that 

1 


I  = 


(54) 


It  is  very  important  to  watch  directions  and  it  is  easy  to 
commit  errors. 

If  each  phase  of  the  star-connected  generator  has  a  resistance 
7*1  and  no  reactance,  then  there  will  be  an  ohmic  drop  ir  in  each 
phase,  and  in  time-phase  with  e,  as  shown  in  Fig.  50.  Then 

E  =  A/3(e  -  ir)  (55) 

in  accordance  with  Kirchhoff's  Second  Law  that  in  any  closed 
circuit  the  algebraic  sum  (vector  sum)  of  the  products  of  the 
current  and  resistance  in  each  of  the  conductors  in  the  circuit  is 
equal  to  the  e.m.f.  in  the  circuit. 

In  a  four-phase  (two-phase)  system,  Fig.  52,  we  have,  if  the 
phase  voltages  between  neutral  and  outside  equal  e, 

E  =  \/2-e  (56) 

for  the  voltages  I-II,  II-III,  III-IV,  and  IV-I. 


THE  ROTATING  FIELD  AND  THE  INDUCTION  MOTOR    71 
The  currents  are 


1 


(57) 


from  Kirchhoff's  First  Law. 

i 


IV 


Four-phase 
System. 


FIG.  52. 


KirchofTs 

First  Law 

Applied  to 

Point  I 


Consider,  next,  a  six-phase  system  (Fig.  53).     The  line  volt- 
ages are 


and  the  currents 


E  =  e 
I  =  i 


(58) 
(59) 


KirchofFs  First  Law 

Six-phase  System  Applied  to  Point  I 

FIG.  53. 

Now,  consider  a  true  poly -phase  network  in  which  there  are  n 
phases  (Fig.  54).     Then  the  time  difference  between  two  phases 

is  — >  and  the  voltage  (with  no  current)  between  phases  is 


In  =  E  =  2e  sin  — 
n 


(60) 


72 


INDUCTION  MOTOR 


Apply  KirchhofFs  First  Law  and  draw  the  current  polygon, 
Fig.  54,  whence,  directly 


I  = 


2  sinl- 


(61) 


The  squirrel-cage  winding  is  such  a  poly-phase  winding  as  Fig. 
55  indicates,  in  which  e  is  the  induced  e.m.f .  per  bar,  e  its  current, 


FIG.  54. — n-phase  system.     Kirchoff's  first  law  applied  to  point  I. 

and  r  its  resistance,  while  E  =  2(e  —  ir)  sin(  — j  is  obviously  the 
ohmic  drop  on  the  two  sections  2R,  we  have 

iR 


E  =  2IR  = 


sin  (— ) 
W 


FIG.  55. — The  squirrel  cage. 
e  =  Voltage  induced  in  bars. 
E  =  Voltage  drop  in  end  ring. 


(62) 


THE  ROTATING  FIELD  AND  THE  INDUCTION  MOTOR    73 

The  end  rings  act,  therefore,  in  such  a  manner  as  to  increase 
the  resistance  r  of  each  individual  bar  by  the  amount 

R 


A  similar  argument  leads  to  a  similar  relation  in  regard  to  the 
reactance  of  the  rings,  but  its  application  is  of  small  importance. 

The  relation  (62)  can  also  be  obtained  from  the  consideration 
that  in  the  squirrel-cage  winding  the  total  loss  is 
Total  Loss  =  n(Pr  +  2PR) 

j  = i 

2  sinf- 


.  • .  Total  Loss  =  m2/  r  +  - \  (63) 

(    2si<); 

from  which  it  follows  that  the  effect  of  the  end  rings  consists 
in  raising  apparently  the  resistance  of  each  bar  by  the  amount 

R  (64) 


as  obtained  before. 

Assuming  again  the  distribution  of  the  flux  belt  in  the  air-gap 
to  follow  a  simple  sine  law,  the  e.m.fs.  and  the  currents  follow 
a  sine  law  distribution  at  any  moment  of  time.  The  e.m.fs. 
and  currents  in  the  end  rings  also  are  in  magnitude  represented 
by  the  ordinates  of  the  curve  E.  The  assumption  of  sine  and 
cosine  curves  implies,  of  course,  the  tacit  assumption  of  an  in- 
finite number  of  phases.  The  relations  obtained  previously  are 
correct  for  any  number  of  phases. 

E.  THE  TORQUE  AND  SLIP  AND  THE  EQUIVALENCE  OF  MOTOR 
AND  TRANSFORMER 

The  theory  of  the  Induction  Motor  is  the  theory  of  the  General 
Alternating-current  Transformer.  Credit  for  this  important 
relation  is  due  to  Dr.  H.  Behn-Eschenburg,  of  Oerlikon,  Switzer- 
land, who  demonstrated  this  relation  in  1893.  It  has  become 
fundamental. 

It  is  evident  that  this  relation  pertains  so  long  as  the  rotor  is 
standing  still.  As  was  shown  by  Mr.  Kapp,  the  theory  of  the  field 


74  INDUCTION  MOTOR 

belt  and  its  interactions,  if  represented  by  vectors,  leads  to  a  polar 
diagram  in  which  time-phase  and  space-phase  are  interchangeable. 
If  the  armature  runs  in  synchronism,  there  is  no  current  induced 
in  the  rotor,  the  no-load  current  corresponding  to  the  open  circuit 
current  of  the  transformer.  If  the  armature  lags  behind  the 
field  in  angular  velocity,  then,  if  ~i  is  the  impressed  frequency, 
and  ^2  the  frequency  corresponding  to  co2,  the  angular  velocity 
of  the  rotor,  the  currents  induced  in  the  rotor  windings  are  of 
frequency  ~i  —  ^2.  If  the  armature  resistance  per  phase  is 
rz)  then  a  current  will  flow 

t,  =  -'  =  2.12  (~'  ~  ~2)  «2F210-'  (65) 


1  7*2 

=  2.12          -^TMO-s  (66) 


The  same  current  will  be  obtained  with  the  secondary  at  rest,  if 

the  external  resistance  is  equal  to  —  •     Therefore,   substitute 

s 

for  the  motor  an  equivalent  transformer  with  a  total  internal 

and  external  resistance  equal  to  R2  =  — 

s 

The  Torque.  —  Imagine  the  rotor  to  be  turned  with  angular 
velocity  (coi  —  o>2)  against  the  magnetic  field,  which  is  supposed 
to  be  at  rest.  If  T  is  the  torque  in  mkg,  then  we  have 

9.81!T-(wi  -  co2)  =  (3^22r2)  (67) 

where  iz  is  the  current,  r2  the  resistance  in  each  phase,  the  rotor 
to  be  assumed  three-phase.     If  it  is  n-phase,  substitute  n  for  3. 

CO    =    27T  —  -  (68) 

p 

where  p  is  the  number  of  north  or  south  poles.     Also 

9.81  -T-ut  =  P  watts. 
From  these  equations  follows 

(69) 

9.81-27r.Tmfca~  -  P  =  3iVr2  (70) 

Pwatta)          (71) 


THE  ROTATING  FIELD  AND  THE  INDUCTION  MOTOR    75 

If  we  want  to  obtain  the  torque  in  foot-pounds,  the  formula  is 

8.5- Tft.  ibs.  =  £^  (3i22r8  +  PwaM4)  (72) 

The  torque  is,  therefore,  proportional  to  the  algebraic  sum  of  the 
output  of  the  motor  plus  the  energy  dissipated  in  the  armature. 
Subtract,  therefore,  the  primary  copper  loss  and  the  core  loss 
from  the  input  as  given  by  the  ordinates  of  the  circle  and  we 
obtain  directly  the  torque  T  in  synchronous  kw. 

The  Slip. — The  slip  of  the  motor  equals  the  loss  in  the  rotor, 
divided  by  the  output  plus  the  loss  in  the  rotor, 


Determination  of  the  Slip 

from  the  Loss  Lines 
FIG.  56. — Torque,  slip,  and  loss  lines  for  the  induction  motor. 

Andre  Blondel1  suggests  a  beautiful  mode  of  representing  the 
slip  by  extending  me,  Fig.  16,  to  an  intersection  with  vector  e\. 
Then  it  can  easily  be  proved  that  the  segment  OX,  X  being  the 
point  of  intersection,  is  a  measure  of  the  slip.  This  method  has 
to  be  modified  if  the  copper  loss  in  the  primary  winding  is  to  be 
taken  into  account  as  diminishing  the  receptive  capacity  of  the 
motor. 

Perhaps  the  simplest  way  to  show  graphically  the  slip  of  the 
motor  is  the  following.  Let  ab  be  the  output  of  the  motor,  be 

be 
the  rotor  loss,  then  slip  =  — . 

Draw  through  g,  the  standstill  position  of  the  motor  in  which 

1  ANDRE  BLONDEL,  "Theories  Graphiques  des  Moteurs  Polyphase^. " 
V Industrie  Electrique,  1896,  p.  77. 


76 


INDUCTION  MOTOR 


the  output  is  zero,  gn  parallel  to  df,  the  primary  copper  loss  line. 
Then 

nh 

o      

ng 

bc:ac::ld:hl 
kl:hl::dl:de::nh  \ng 
bc:ac::nh:ng q.e.d.         (74) 


+  80  %  +  00  %  -HO  %  +  20  %  NJ Q       -  20  %  -  40 %  -  60 %  -80  %  -100  %  Slip 


X)  K.W. 


-100 


Fig.  57. — Torque  as  a  function  of  rotor  resistance,  slip,  and  primary  input. 

Divide  ng  into  a  percentage  scale  and  the  slip  may  be  read  off  for 
each  current.     At  g  the  slip  is  100  per  cent,  at  n  it  is  zero. 

Torque  Curves. — The  results  obtained  in  polar  coordinates 
shall  .now  be  represented  in  Cartesian  coordinates.  We  shall 
first  use  as  abscissa  the  watts  input  of  the  motor,  and  secondly 
the  slip,  representing  all  characteristics  as  functions  of  these 
two  parameters  (Fig.  57). 


THE  ROTATING  FIELD  AND   THE  INDUCTION  MOTOR    77 

It  is  interesting  to  note,  and  obvious  from  the  diagrams,  that 
there  is  a  maximum  torque  for  a  given  motor  frame.  This  torque 
may  occur  at  starting  or  at  any  speed.  It  may  be  varied  by  the 
resistance  of  the  rotor,  thus  making  it  possible  to  start  with  maxi- 
mum torque  by  increasing  the  rotor  resistance.  The  reader  may 
be  trusted  to  draw  many  other  instructive  conclusions  from  the 
diagrams.  Torque  may  be  measured  conveniently  in  ' '  synchro- 
nous kw.,"  which  is  the  kw.  at  angular  velocity  coi  corresponding 
to  the  torque  at  angular  velocity,  o>2. 

F.  HIGHER  HARMONICS  IN  THE  FIELD  BELT  AND  THEIR  EFFECT 
UPON  THE  TORQUE 

Examining  Fig.  41  it  is  noticed  that  the  field  belt  does  not 
have  the  form  of  a  sine  wave.  It  consists,  therefore,  of  a  funda- 
mental sine  wave  and  of  higher  harmonics  of  this  fundamental. 

To  study  these  effects,  consider  first  a  quarter-phase  motor, 


3d  Harmonic 

Phase. I  Lead!  .  Phase -II 
Phase  IH-Lagi.  Phase  II- 1 
. '.    IH  and  II-II  Rotate 
Backward! 


5th  Harmonic 
Phase  I  Leads  -  Phase  II 
Phase  IH  Leads-  Phase  II- 
IH  and  II-E  Rotate 
Forwards 


7th  Harmonic 
rbase  I  Leads  •  Phase  IX 
Phase  IH  Lags-  Phase  II-H 
. '.   IH  and  II  H  Rotate 
Backwards 


FIG.  58. — Harmonics  in  the  field  belt  of  a  two-phase  motor. 


78 


INDUCTION  MOTOR 


important  application  of  which  has  recently  been  made  to  the 
U.  S.  Battleship  New  Mexico,  as  discussed  fully  in  Chap.  VIII. 
The  third  harmonic  combines  in  the  two  phases  in  such  a  manner 
that,  if  Phase  II  Fundamental  is  behind  Phase  I  Fundamental, 
then,  .Phase  II  3d  Harmonic  is  ahead  of  Phase  I  3d  Harmonic 
(Fig.  58).  Hence,  the  third  harmonic  flux  belt  in  a  quarter- 
phase  motor  produces  a  backward  torque. 

It  must  be  emphasized  that  we  are  talking  about  harmonics  in 
the  flux  belt  and  not  in  the  e.m.f.  of  the  supply  circuit.     The 


Third.  Harmonic 
Torque 


FIG.  59. — The  effect  of  the  third  harmonic  in  the  field  belt  on  the  torque  of  the 

two-phase  motor. 

effects  of  the  latter  may  be  neglected  as  a  little  thought  indicates. 
The  effects  of  the  harmonics  in  the  flux  belt,  however,  consist  in 
setting  up  rotating  fields  with  angular  velocity  corresponding  to 
the  fundamental  supply  frequency,  whose  effect  is  therefore  the 
same  as  the  superimposition  of  3,  5  or  7  times  the  number  of 
poles  would  have  upon  the  main  fundamental  flux  belt. 

We  have  already  seen  that  in  a  two-phase  motor  the  third 
harmonic  acts  as  a  brake  (Fig.  59).  In  a  two-pole  motor  fed 
from  25  cycles  the  synchronism  of  the  fundamental  is  1,500 


THE  ROTATING  FIELD  AND  THE  INDUCTION  MOTOR    79 

r.p.m.,  while  the  synchronism  of  the  third  harmonic  takes  place 
at  —500  r.p.m.,  but  the  field  rotating  backwards  will  produce  a 
torque  curve  as  indicated  in  Fig.  59.  It  is  quite  evident  from 
Fig.  59,  that  such  backward  torque  may  be  extremely  serious  as 
it  diminishes  the  starting  torque.  If  a  squirrel-cage  rotor  is 
used,  the  motor  may  be  unable  even  to  start  at  all. 


: Slip 

Wave 

from  Fundamental  "Wave 
Harmonic  Belts 


I  Torque 

II  Torque  Resulting 
and  3d,  5th,  &  7th 


FIG.  60. — The  effects  of  harmonics  in  the  field  belt  on  the  torque  of  a  two-phase 

motor. 

Drawing  the  phases  for  the  fifth  harmonic,  it  will  be  seen  that 
in  the  quarter-phase  motor  the  fifth  harmonic  produces  a  for- 
ward torque  whose  synchronism  occurs  at  +300  r.p.m.  The 
seventh  harmonic  gives  a  backward  torque  whose  synchronism 
occurs  at  —215  r.p.m.  These  three  harmonics  have  been  drawn 
into  Fig.  60  showing  the  resultant  torque.  The  dead  points, 
which  were  so  often  observed  in  two-phase  motors  during  the 
development  stages,  are  particularly  interesting. 


80 


INDUCTION  MOTOR 


An  examination  of  the  effects  of  the  third  harmonic  in  a  three- 
phase  system,  if  the  sine  waves  are  drawn,  shows  that  the  third 
harmonics  of  the  three  circuits  are  in  phase  with  each  other, 
therefore,  they  produce  no  rotating  field.  Their  effect  upon  the 
torque  is  therefore  that  of  a  single-phase  induction  motor  having 
three  times  the  number  of  poles.  Its  torque  is  shown  in  Fig.  61 


5thH 


Slip 


3dH 


FIG.  61.— The  effect  of  a  third  harmonic  in  the  field  belt  on  the  torque  of  a 
three-phase  motor.  The  torque  for  the  5th  harmonic  is  shown  but  not  added 
to  the  fundamental  torque. 

where  also  the  torques  of  the  fifth  and  seventh  harmonics  are  indi- 
cated. The  fifth  harmonic  gives  backward  torque,  the  seventh 
harmonic  gives  a  forward  torque.  The  torque  curve  of  the  single- 
phase  induction  motor  will  be  treated  at  length  inChap.  XVII.1 

1  This  entire  subject  of  the  effects  of  higher  harmonics,  both  in  the  field 
belt  and  in  the  supply  circuits,  was  treated  brilliantly  by  ANDR£  BLONDEL,  as 
early  as  1895,  in  his  paper,  -''Quelques  Proprie"tes  Generates  des  Champs 
Magne*tiques  Tournants,"  L'Eclairage  Electrique,  Paris,  to  which  classic 
and  fundamental  paper  all  readers  should  refer. 


THE  ROTATING  FIELD  AND  THE  INDUCTION  MOTOR    81 
G.  EXPERIMENTAL  DATA 

A  vast  amount  of  experimental  material  has  been  accumulated 
since  the  first  publication  of  this  theory  25  years  ago.  The 
experimental  proof  on  induction  motors  of  the  circle  character- 
istic was  already  given  in  my  paper  of  Jan.  30,  1896,  on  one 
machine  only.  Literally,  hundreds  of  thousands  of  motors, 
aggregating  millions  of  horsepower  have  been  tested  since 
according  to  this  method  and  the  theory  is  now  solidly  estab- 
lished. In  subsequent  chapters  numerous  experimental  data 
are  given  so  that  we  shall  not  here  refer  to  the  subject  any  more. 

H.  COLLECTION  OF  DATA 

The  most  important  data  in  the  design  of  the  three-phase  induc- 
tion motor  are  here  collected  together: 

Leakage  Factor: 

a  =  ViVZ  —  1  (14) 

Maximum  Power  Factor: 


Magnetic  Flux: 

0i  =  2.12~-zr/<YlO-8  volts  (49) 

Magnetizing  Current: 

t^w  ^ 

Slip: 

COi    —    C02  OlZ'2  /**n\ 

(73) 


3*22r2  +  P 
Torque: 

61.6  Dmkg  =  ^-  [3*2V2  +  Pwatt8]  (71) 

Input: 

Pi  =  3erfi  cos  4/1  (74) 

Output: 

Pwatts  =  3ei*i  cos  ^t  -  SiiVi  -  F  -  Q  (75) 

where  F  =  friction  loss  in  watts,  and  Q  the  loss  through  hysteresis 
and  eddy  currents. 

Efficiency: 

(76) 


cos 


CHAPTER  VI 

THE  INDUCTION  GENERATOR 
A.  THE  THEORY  OF  TORQUE  AND  SLIP 

If  an  induction  motor  is  driven  by  an  external  torque  applied 
to  its  rotor  at  a  speed  above  synchronism,  in  the  direction  of  its 
rotation  as  a  motor,  its  slip  becomes  negative  as  co2  becomes 
greater  than  coi.-  As  the  supply  circuit  continues  to  impress 
upon  the  stator  a  rotating  field  through  which  the  rotor  con- 
ductors cut  at  angular  velocity  coi  —  co2,  it  is  .apparent  that  elec- 
tric energy  will  be  impressed  upon  the  supply  circuit.  The 
mechanical  input  required  to  turn  the  rotor  is  equal  to  the 
electrical  output  plus  the  losses  in  the  rotor  and  stator,  exactly 
as  in  the  case  of  a  synchronous  generator.  An  induction  machine 
operating  above  synchronism  is  therefore  called  an  Induction 
Generator. 

A  careful  consideration  of  all  that  has  been  said  on  the  subject 
of  the  induction  motor  makes  it  clear  that,  on  account  of  the 
negative  slip,  the  secondary  current  will  be  generator  current 
instead  of  motor  current.  Its  vector  will  be  "downwards" 
from  F2,  instead  of  "upwards"  as  in  the  motor.  If  we  neglect 
primary  resistance  the  semi-circle  below  the  abscissa  is  the  locus 
of  the  primary  current.  If  we  take  into  account  the  primary 
resistance  the  complete  circle  is  also  swept  out,  but  now  divided 
differently  into  a  motor  range  and  a  generator  range. 

The  behavior  of  such  an  induction  generator  is  indeed  curious. 
Prof.  M.  I.  Pupin1  talks  of  " negative  resistance"  in  this  case. 
It  is  quite  unnecessary  to  make  a  complicated  subject  any  more 
complicated  by  mystery.  It  makes  little  difference  how  the 
field  is  supplied,  whether  from  the  external  circuit  or  by  some 
other  contrivance.  And  herein  seems  to  lie  the  difficulty  of 
understanding  the  action  of  this  machine.  The  effect  of  primary 
resistance  in  lowering  the  effective  part  of  the  impressed  voltage 
is  also  worth  pondering.  As  one  never  understands  anything 
which  one  has  not  thought  out  for  oneself,  we  shall  leave  these 
matters  to  the  reader. 

1  Transactions  A.  I.  E.  E.,  1918,  Part  I.,  p.  685. 

82 


THE  INDUCTION  GENERATOR 


83 


The  slip  is  again  expressed  by  the  loss  in  the  rotor  divided  by 
the  electrical  output  plus  the  rotor  loss.  The  scale  of  the  slip, 
extended  to  the  left,  is  therefore  a  measure  of  the  negative  slip 
of  the  induction  generator  (Fig.  62). 

The  torque  is  measured  directly  between  the  circle  and  the  pri- 
mary copper  loss  line.  The  results  are  also  shown  in  Cartesian 
coordinates  in  Fig.  63. 


/A.B=Motor  Torque 
CZ?=Generator  Torque 
OB  =  Motor  Current 
OD=Generator  Current 
JkTG=Motor  Slip 
M.H=Generator  Slip 

FIG.  62, — The  torque  and  the  slip  in  the  induction  generator. 

B.  STABILITY 

If  the  torque  which  it  is  required  to  start  by  an  induction  motor, 
see  Fig.  63,  with  short-circuited  secondary,  is  ab,  and  if  this  torque 
increases  according  to  the  line  6A,  then  the  motor  rapidly  accel- 
erates and  operates  at  the  point  1  in  a  stable  condition.  If  the 
torque  curve  of  the  load  follows  bB,  then  if  point  N  can  be  passed, 
a  stable  point  will  be  reached  at  2.  If  the  torque  curve  follows 
the  line  bC  then  the  motor  will  "stick"  at  P,  and  it  will  require 
the  application  of  external  torque  to  make  it  reach  Q,  from  where 
the  rotor  will  accelerate  and  operate  stably,  at  least,  theoreti- 
cally, at  point  3. 

When  operating  as  an  induction  generator  a  short-circuit  of 
the  line  removes  the  excitation  of  the  induction  generator  and 
its  prime  mover  accelerates  to  the  runaway  point.  It  is  also 


84 


INDUCTION  MOTOR 


-.25    -.50    -.75    -l.JXK-1.25  -1.50  -2.00 

GENERA 


Slip     -H        .75      .50      .25      0 
MOTOR 


FIG.  63. — Stability  in  the  induction  motor  and  in  the  induction  generator 


0      10 20      0     40     50     30    70  .  80     00     1QQ  =Amperei 


FIG.  64. — The  experimental  circle  diagram  for  an  induction  machine  acting  as 
motor  and  generator. 


THE  INDUCTION  GENERATOR  85 

essential  that  the  characteristic  torque  of  the  prime  mover  shall 
not  increase  with  increasing  speed.  Water  wheels  usually  lose 
their  torque  at  approximately  twice  the  normal  speed,  therefore, 
such  a  torque  characteristic  of  the  prime  mover  as  indicated  by 
line  Z),  would  establish  a  stability  point  at  4.  A  sudden  lowering 
of  the  supply  voltage  may,  however,  cause  serious  trouble,  in 
view  of  the  sensitiveness  of  the  induction  machine  in  regard  to 
its  primary  impressed  voltage,  a  drop  of  10  per  cent  reducing 
the  torque  20  per  cent,  and  point  5  may  be  passed  and  the  prime 
mover  and  generator  run  away. 

C.  EXPERIMENTAL  DATA 

Figure  64  shows  a  carefully  obtained  test  on  an  induction  ma- 
chine over  the  entire  range  of  operation,  made  in  1904,  by  the 
writer  in  his  testing  department  in  South  Norwood,  Ohio. 


CHAPTER  VII 

THE    SHORT-CIRCUIT   CURRENT   AND   THE   LEAKAGE 

FACTOR 

A  great  aid  in  the  design  of  poly-phase  induction  motors  is 
afforded  by  the  short-circuit  characteristic.  Attention  was  first 
called  to  this  in  1893  by  Dr.  H.  Behn-Eschenburg  and  Mr.  Gisbert 
Kapp.  In  a  motor  having  a  squirrel-cage  armature,  the  start- 
ing current  under  different  voltages  is  identical  with  the  short- 
circuit  characteristic.  If  the  resistances  of  the  armature  and 
of  the  field  were  known,  the  power  factor  of  the  supply  current 
could  be  calculated;  thus  it  would  not  even  be  necessary  to  use  a 
wattmeter  unless  great  accuracy  was  required.  Theoretically, 
then,  the  short-circuit  characteristic  is  sufficient  for  the  deter- 
mination of  the  leakage  factor,  if  the  magnetizing  current  is 
known.  In  practice,  however,  it  is  inadvisable,  in  the  majority 
of  cases,  to  depend  upon  the  short-circuit  curve,  on  account  of 
the  corrections  which  become  necessary.  If  the  total  resistance 
of  the  motor  is  small,  the  lag  of  the  current  amounts  to  nearly  a 
quarter  of  a  period.  The  inductance  of  a  motor  at  standstill 
should  always  be  as  small  as  possible,  therefore,  a  very  large 
current  will  go  through  the  motor  at  full  voltage.  Now,  the  leak- 
age factor  is  more  or  less  dependent  upon  the  intensity  of  the 
currents  which  cause  the  leakage,  hence,  the  leakage  factor  at 
starting  with  only  a  small  resistance  in  the  armature,  may  be 
very  different  from — and  as  a  rule  it  is  smaller  than — the  leakage 
factor  upon  which  the  diagram  is  based.  In  fact,  we  do  not 
work  the  motor  in  that  quadrant  of  the  circle  which  corresponds 
to  the  short-circuit  characteristic.  If,  therefore,  I  shall  not  make 
much  use  of  the  short-circuit  curve  for  the  determination  of  the 
absolute  value  of  the  leakage  factor,  I  shall  all  the  more  avail 
myself  of  it  for  comparative  purposes,  where  it  does  not  matter  so 
much  whether  the  absolute  value  is  correct  or  not,  as  the  relative 
value  is  chiefly  of  importance.  Such  questions  as  the  influence 
of  a  closed  or  open  slot,  of  the  number  of  slots,  of  the  air-gap, 
and  of  the  pole-pitch  upon  the  leakage  factor,  can  all  be  answered 
by  consulting  the  short-circuit  characteristics.  I  shall  proceed 

86 


SHORT-CIRCUIT  CURRENT  AND  LEAKAGE  FACTOR       87 


to  discuss,  point  for  point,  the  influence  of  these  factors  on  the 
leakage  coefficient. 

A.  THE  SLOTS 

The  curves  A  and  B,  in  Fig.  65,  represent  the  short-circuit 
characteristics  for  a  closed  slot  of  the  shape  marked  A,  and  of  the 
open  slots  of  the  shape  marked  B.  The  slots  of  the  rotor  were 
closed  in  each  case.  Curve  C  shows  the  ideal  short-circuit  curve 
obtainable  only  if  the  leakage  paths  contain  no  iron  at  all. 
Curve  D  is  the  magnetizing  current  reduced  from  the  measured 
value  of  42.2  amperes  at  1,900  volts  to  the  various  voltages  of  the 


120 


100 


40 


20 


100       200       300       400       500       600       700       800       900 

Volts 
FIG.  65. — Short  circuit  characteristics  for  closed  and  open  slots. 

diagram.  The  increase  of  the  magnetic  reluctance  of  the  main 
field  through  the  opening  of  the  slots  proved  too  small  to  influence 
the  magnetizing  current  in  any  perceptible  manner. 

We  see  that  for  voltages  above  600  the  curves  A,  B,  and  C 
converge;  in  other  words,  the  short-circuit  current  at  the  full 
voltage  of  1,900  will  be  almost  the  same  whether  the  slots  are 
open  or  closed.  Hence,  the  maximum  power  input,  which  can 
be  impressed  upon  the  motor,  is  by  no  means  so  dependent 
upon  the  form  of  the  slot  as  is  usually  assumed.  The  tendency 
of  the  closed  slot  is  to  change  the  fundamental  diagram  in  the 
way  shown  by  the  full  line  curve  in  Fig.  65.  From  this  curve  it 


88  INDUCTION  MOTOR 

follows  that,  though  the  maximum  power  factor  is  slightly  re- 
duced, yet  the  maximum  output  of  the  motor  is  hardly  affected 
as  is  indicated  by  the  approximate  equality  of  the  maximum  or- 
dinates  of  the  heavy  line  and  dotted  curves  which  measure  the 
maximum  power  receptivity  of  the  motor.  If  the  iron  bridges 
are  kept  very  thin,  excellent  motors  can  be  built  with  closed 
slots.  The  objection  to  closed  slots  is  a  commercial  one,  the 
high  cost  of  labor  of  winding  the  coils  within  the  machine  instead 
of  on  formers  outside  the  machine.  During  the  development  pe- 
riod of  the  induction  motor,  25  to  30  years  ago,  labor  cost  was 
less  of  a  telling  factor  and  motors  built  in  the  pioneer  factories  of 
Oerlikon  and  of  Brown,  Boveri,  and  Company ,  utilizing  the  skilled 
Swiss  labor  then  available,  were  ordinarily  supplied  with  closed 


FIG.  66. — The  effect  of  saturation  of  the  leakage  path  upon  the  circle  diagram. 
(From  the  1st  edition  of  1900.) 

slots  and  hand-wound  coils.  Needless  to  say,  times  have 
changed,  skilled  and  inexpensive  labor  has  departed,  and  cheap 
commercial  methods — fool-proof  methods  which  enable  the 
manufacture  of  coils  to  be  made  by  inexperienced  hands — have 
supplanted  the  early  notions.  To  such  processes  we  apply  the 
term  of  "  progress,"  though  it  would  perhaps  be  nearer  the  facts 
to  say  that  evolution,  far  from  meaning  progress,  is  solely  the 
adaptation  to  new  conditions  and  environment.  The  survival  of 
the  fittest  means,  to  be  sure,  the  fittest  in  regard  to  its  surroud- 
ings,  the  best  adapted  to  these  surroundings,  not  by  any  means, 
however,  the  "best"  in  any  ethical  sense.  Huxley  deplored 
that  " fittest"  had  the  connotation  of  "best"  and  was  therefore 
a  term  unhappily  chosen  by  Herbert  Spencer.  A  design  is  a 
compromise,  and  the  best  design  is  that  which  combines  the 
greatest  number  of  advantages  with  the  least  number  of  draw- 
backs. As  an  old  French  proverb  has  it,  our  choice  lies  between 
the  bad  and  the  worse,  and  not  between  the  bad  and  the  good. 


SHORT-CIRCUIT  CURRENT  AND  LEAKAGE  FACTOR       89 

B.  NUMBER  OF  SLOTS  PER  POLE 

At  the  outset  we  wish  to  emphasize  that  we  always  consider  one 
closed  magnetic  circuit  consisting  of  a  pair  of  poles.  We  trust 
the  reader  can  apply  the  arguments  readily  to  the  particular 
multi-polar  case,  if  he  understands  what  goes  on  in  one  pair  of 
poles. 

Theoretically,  we  should  have  as  many  slots  as  possible, 
commercially,  it  is  advisable  to  have  as  few  as  possible.  Either 
extreme  is  impracticable. 

In  a  general  way,  the  influence  of  the  number  of  slots  upon 
the  leakage  can  be  seen.  The  more  conductors  we  have  in  a 
slot  the  larger  will  be  the  leakage  field  surrounding  the  slot. 
With  enough  accuracy  for  our  present  consideration,  the  active 
field  is  the  same  whether  we  distribute  the  same  number  of  con- 
ductors in  a  few  slots  or  in  many.  The  e.m.f.  induced  by  this 
field  may,  therefore,  be  assumed  constant  independent  of  the 
number  of  slots.  To  fix  ideas,  let  us  take  a  specific  case.  If  we 
have,  for  instance,  100  conductors  arranged  in  five  slots,  there  are 
20  conductors  in  each  slot.  The  leakage  field  per  slot  is  produced 
by  these  20  conductors,  it  is  therefore  proportional  to  their 
number.  The  e.m.f.  induced  by  the  leakage  field  per  slot  in  the 
20  conductors  in  each  slot,  is  proportional  to  20  X  20.  Hence, 
the  total  leakage  e.m.f.  induced  in  the  100  conductors  by  the 
five  leakage  fields  is  proportional  to  20  X  20  X  5  =  2,000. 

Now,  let  us  arrange  the  100  conductors  in  10  slots.  Each 
slot,  then,  contains  10  conductors,  the  leakage  field  per  slot  being 
proportional  to  10.  The  e.m.f.  induced  by  the  leakage  field  in 
the  10  conductors  in  each  slot  is  proportional  to  10  X  10. 
Hence  the  total  e.m.f.  induced  in  all  the  conductors  in  the  10 
slots  is  proportional  to  10  X  10  X  10  =  1,000.  In  other  words, 
the  e.m.f.  of  the  leakage  is  twice  as  large  in  the  case  of  five  slots 
as  in  the  case  of  10  slots. 

The  above  consideration  assumes  equal  leakage  reluctance  in 
the  two  cases.  The  argument  gives  a  general  conception  only. 

C.  CHARACTERISTICS  OF  ROTOR  WINDINGS 

To  study  the  effect  of  the  number  of  phases  in  the  rotor  and 
also  the  effect  of  the  closeness  of  the  rotor  conductors  to  the 
air-gap,  a  series  of  tests  was  made  as  follows: 


90 


INDUCTION  MOTOR 


A.  Rotor  three-phase  star-connected. 

B.  Rotor  removed  from  stator.     Leakage  field  closed  through  air. 

C.  Rotor  squirrel-cage,  two  conductors  in  bottoms  of  slots. 

D.  Rotor  squirrel-cage,  two  conductors  close  to  top  of  slot. 

E.  Rotor  squirrel-cage,  one  conductor  close  to  top  of  slot. 

The  results  are  plotted  in  Fig.  67,  showing  the  inferiority  of 
the  three-phase  rotor  to  the  squirrel-cage  so  far  as  leakage  is 


40  50          60 

Amperes 

FIG.  67. — Short-circuit  current  of  induction  motor  with  (a)  Y-connected  rotor, 
(b)  squirrel  cage  rotor,  (c)  rotor  conductors  near  the  gap,  (d)  rotor  conductors 
at  bottom  of  slot,  (e)  no  rotor  at  all. 


concerned  and  the  advantage  obtained  by  bringing  the  conductors 
close  to  the  gap.  Reducing  the  section  of  the  conductors  to  one- 
half,  Case  "C, "  does  not  seem  to  constitute  a  noticeable  gain. 

It  is  interesting  to  note  the  relative  independence  of  the  magni- 
tude of  the  leakage  field  in  respect  to  the  air-gap,  as  without  a 
rotor  inside  the  stator  this  leakage  reluctance  is  only  slightly 
smaller — considering  both  primary  and  secondary  leakage 
reluctances  in  parallel — than  with  the  rotor  in  position. 


SHORT-CIRCUIT  CURRENT  AND  LEAKAGE  FACTOR       91 
D.  TEST  DATA 

To  enable  the  reader  to  form  for  himself  an  opinion  of  the 
accuracy  of  the  theory,  I  give  the  complete  experimental  data  of 
a  small  three-phase  current  motor  with  short-circuited  armature. 
I  am  taking  an  old  motor  designed  25  years  ago. 

The  motor  shall  develop  20  hp.,  at  a  voltage  of  380  between 
the  lines,  and  a  frequency  of  47  p.p.s  The  slots  in  armature 
and  field  were  closed,  but  the  bridges  were  thin.  The  shape  of 
the  slots  in  armature  and  field  is  represented  in  Fig.  68.  The 
motor  had  six  poles,  therefore  its  synchronous  speed  was  940  r.p.m. 


FIG.  68. — Slots  of  stator  and  rotor  of  20  hp.  six-pole  motor,  designed  in  1896. 

The  following  table  shows  the  starting  or  short-circuit  current 
as  a  function  of  the  terminal  volts  measured  between  the  lines: 


Volts 

Amperes 

Frequency 

Volts 

Amperes 

Frequency 

82 

11 

46.7 

200 

56 

46.7 

108 

20 

233 

70 

.... 

135 

30 

.... 

273 

90 

.... 

162 

40 

314 

112 

These  values  are  graphically  represented  in  Fig.  69.  From 
this  curve  we  interpolate  for  380  volts  a  current  of  140  amperes. 
The  energy  dissipated  into  heat  in  the  windings  and  in  the  iron 
amounted  to  30  kw.  This  point  lies  upon  the  broken  line  curve 
in  Fig.  71,  deviating,  therefore,  not  inconsiderably  from  the  full 
line  curve,  representing  the  semi-circle  of  our  diagram.  The 
data  determining  this  circle  are  given  in  the  following  table: 


92 


INDUCTION  MOTOR 


2 
2 

fa 

1 

+3 

I 

a 

4* 

o 

Effi- 

Power 

Apparent 
effi- 

S 
ex 
tf 

§  4 
I 

1 

1 

•9 

3 
1 

ciency, 
per  cent 

factor, 
per  cent 

ciency, 
per  cent 

930 

932 

380 

5.5 

2,100 

1,300 

62.0 

58.0 

36.0 

931 

940 

382 

9.0 

5,400 

4,050 

75.0 

90.7 

68.0 

926 

930 

380 

14.5 

9,000 

7,300 

81.0 

94.4 

76.5 

910 

918 

380 

20.5 

12,900 

10,800 

84.0 

95.5 

80.2 

912 

925 

380 

26.0 

15,600 

13,200 

84.5 

91.5 

77.3 

894 

912 

380 

31.0 

18,600 

15  ,  600 

84.0 

91.0 

76.5 

892 

922 

380 

42.0 

25  ,  100 

20,600 

82.0 

91.0 

74.5 

880 

922 

380 

57.0 

32,400 

24,300 

75.0 

86.5 

65.0 

846 

896 

382 

64.5 

35,400 

25  ,  500 

72.0 

83.0 

59.7 

860 

940 

391 

74.5 

40,100 

26,100 

65.0 

79.5 

51.7 

1GO 


140 


120 


100 


40 


20 


Starting  Curren 


7 


100 


200 
Volts 


400 


FIG.  69. — The   short-circuit   current  of   20   hp.    induction   motor   with   closed 
slots.     (Design  of  1896.) 

The  curves  corresponding  to  these  data  are  graphically  repre- 
sented in  Fig.  70.  The  maximum  output  of  the  motor  is  35  hp. 
This  point  lies  very  near  the  maximum  ordinate  of  the  semi- 


SHORT-CIRCUIT  CURRENT  AND  LEAKAGE  FACTOR       93 

circle  in  Fig.  71.     The  diameter  of  this  circle  is  122.5  amp.  the 
magnetizing  current  4.5  amp.  hence 

'  -     -  °-°367 


The  maximum  power  factor  attainable  is,  equation  (19), 

1 


COS  $0    = 


2(7 


0.93 


0  10  20  30  40  45 

Kilowatts 

FIG.  70. — Characteristic  data  of  20  hp.  induction  motor  from  tests  made  in  1896. 


Figure  71  and  the  data  given  above  clearly  show  the  inaccuracy 
which  would  arise  if  we  were  to  use  the  short-circuit  current  as  a 
means  of  determining  the  absolute  value  of  the  leakage  factor. 

I  want  to  call  attention  to  the  load  losses,  which  are  always 
present  in  poly-phase  motors,  the  causes  of  which  are,  however, 
still  very  obscure. 

The  maximum  efficiency  of  this  motor  is  84.5  per  cent.  The 
losses  are : 


94  INDUCTION  MOTOR 


Hysteresis,  eddy  currents  and  friction 

Ohmic  loss  in  primary 

Ohmic  loss  in  secondary 


Watts 

800 
600 
200 


Total  losses 1 , 600 


Output 13.200 


Input . 


14,800 


FIG.  71. — The  circle  diagram  of  20  hp.  induction  motor,  taken  in  1896. 

The  energy  which  the  motor  actually  consumed  amounted  to 
15,600  watts,  corresponding  to  an  additional  loss — the  load  loss — 
of  6  per  cent  of  the  output.  The  load  loss  increases  rapidly 
with  increasing  load,  until  it  becomes  equal  to  all  other  losses 
taken  together. 

Opening  the  slots  has  a  decided  tendency  to  diminish  the  load 
loss  considerably,  therefore,  it  is  probable  that  the  seat  of  this 
waste  of  energy  is  in  the  bridges. 

E.  THE  LEAKAGE  FACTOR 

There  are  two  questions  of  the  most  vital  interest  which  intrude 
themselves  at  every  step  upon  the  designer.  The  first  is:  "How 
does  the  maximum  output  a  poly-phase  motor  is  capable  of 
yielding  depend  upon  the  length  of  the  air-gap?"  In  other 
words,  if  the  air-gap  is  increased,  does  this  to  any  great  extent 
decrease  the  output?  And  does  a  decreased  air-gap  increase  the 
output  of  the  motor?  The  second  question  is  this :  If  a  motor  is 
wound  for  four  poles,  and  we  want  to  wind  it  for  eight  poles, 


SHORT-CIRCUIT  CURRENT  AND  LEAKAGE  FACTOR       95 


provided  the  frequency  and  the  induction  in  the  air-gap  remain 
the  same,  does  the  output  decrease  in  the  ratio  of  4  -r-  8,  or,  what 
relation  exists  between  the  maximum  output  of  the  motor  and 
the  number  of  poles?  I  shall  now  proceed  to  answer  these 
questions. 

F.  THE  INFLUENCE  OF  THE  AIR-GAP  UPON  THE  LEAKAGE  FACTOR 

In  order  to  determine  the  interdependence  between  the  mag- 
netic reluctance  of  the  main  field  and  the  leakage  factor,  the  fol- 
lowing experiment  was  made :  The  magnetic  field — the  stator — of 
a  three-phase  current  motor  was  provided  with  two  armatures, 
the  diameters  of  which  were  so  chosen  as  to  create  an  air-gap  of 
0.5  mm.  and  one  of  1.5  mm.  The  magnetizing  currents  were 
then  measured  as  well  as  the  short-circuit  currents.  The  follow- 
ing tables  contain  the  results  of  the  experiment : 

MAGNETIZINCf    CURRENTS    AT    50~ 


A  =  0.5  mm. 

A  =  1.5  mm. 

Volts 

Amperes 

Volts 

Amperes 

37.0 

1.00 

16.5 

1.05 

56.5 

1.50 

34.7 

2.25 

78.0 

2.10 

55.5 

3.40 

95.5 

2.55 

75.0 

4.70 

116.0 

3.20 

77.0 

4.80 

.... 

77.5 

5.00 

.... 

99.0 

6.40 

.... 

113.0 

7.30 

114.5 

7.40 

SHORT-CIRCUIT  CURRENTS  AT  50' 


A  =  0.5  mm. 


A  —   1.5  mm. 


Volts 

Amperes 

Volts 

Amperes 

17.4 

5.1 

16.2 

5.40 

36.0 

12.5 

36.0 

14.85 

36.5 

12.7 

37.6 

14.00 

58.0 

20.3 

55.5 

22.50 

77.0 

29.5 

79.0 

33.50 

Resistance  of  primary:  Each  phase,  0.05  ohm. 
Resistance  of  secondary:  Each  phase,  0.50  ohm. 


96 


INDUCTION  MOTOR 


The  data  of  these  tables  are  graphically  represented  in  Figs.  72 
and  73.  The  most  remarkable  fact  illustrated  by  these  curves 
is  that  the  short-circuit  current  remained  almost  unaltered  for 
the  two  different  air-gaps.  We  interpolate  from  these  curves 
for  110  volts  in  each  phase  the  respective  short-circuit  and 
magnetizing  currents,  and  thus  get  the  following  values: 


40 


30 


20 


10 


20 


40 


100 


120 


=  0.6  m.m. 


60 
Volts 

I  Magnetizing  Current 

II  Short-circuit  Current 

FIG.  72. — The  leakage  factor  of  the  induction  motor.     The  effect  of  the  air-gap 
on  the  leakage  factor.     Air-gap  A  =  .50  mm. 

A  =  0.5  mm. 

Magnetizing  current:  3  amp. 

Short-circuit  current:  42  amp. 

Watts  consumed:  8,550  watts. 

A  =  1.5  mm. 

Magnetizing  current:  7  amp. 

Short-circuit  current:  47.5  amp. 

Watts  consumed:  9,100  watts. 


SHORT-CIRCUIT  CURRENT  AND  LEAKAGE  FACTOR      97 

With  these  data  the  semi-circles  in  Fig.  74  are  drawn.     We  thus 
find  for  the  leakage  factor  the  following  values: 


A  =  0.5  mm. 
A  =  1.5  mm. 


a-  =  0.058 
a-  =  0.128 


The  leakage  factor  is,  according  to  these  experiments,  directly 
proportional  to  the  magnetizing  current,  or,  in  other  words,  to 


40 


30 


20 


10 


50' 


Volts 

Magnetizing  Current 
Short-circuit  Current 


100  120 

=  1.5  m.m. 


FIG.  73. — The  leakage  factor  of  the  induction  motor.     The  effect  of  the  air-gap 
on  the  leakage  factor.     Air-gap  A  =  1.50  mm. 

the  magnetic  reluctance  of  the  main  field.  In  our  case  the 
reluctance  of  the  iron  path  is  not  negligible,  as  the  air-gap  of 
0.5  mm.  is  very  small,  otherwise  the  leakage  factor  would  have 
been  proportional  to  the  air-gap. 

Now,  this  result  is  highly  interesting.     As  a  glance  at  Fig.  74 


98 


INDUCTION  MOTOR 


shows,  the  energy  which  the  motor  is  capable  of  taking  in  at  the 
voltage  of  110  is  the  same  whether  the  air-gap  is  small  or  large. 
Hence,  the  overload  that  a  motor  is  able  to  stand  is  independent  of 
the  air-gap. 


FIG    74. — The  leakage  factor  of  the  induction  motor.     The  circle  diagram  for 

different  air-gaps. 


This,  of  course,  holds  good  only  of  small  air-gaps. 
The  air-gap  influences  merely  the  strength  of  the  magnetizing 
current,  but  not  the  output. 


O 


FIG.  75. — The  leakage  factor  of  the  induction  motor.     The  shape  of  the  slots 
in  the  experimental  motor.     (1896.) 


The  curves  of  the  short-circuit  currents  in  Figs.  72  and  73  are 
almost  straight  lines  owing  to  the  open  slots  in  the  field  of  our 
motor. 


SHORT-CIRCUIT  CURRENT  AND  LEAKAGE  FACTOR       99 

There  remains  to  be  answered  the  second  question:  How  is 
the  leakage  factor  dependent  upon  the  pitch  of  the  poles? 

G.  THE   INFLUENCE   OF   THE  POLE-PITCH  UPON  THE  LEAKAGE 

FACTOR 

Before  entering  upon  the  experiments  made  to  clear  up  this 
point,  I  shall  attempt  to  show  deductively  how  the  leakage  factor 
may  be  expected  to  vary  with  the  pole-pitch. 

Figure  75  gives  a  view  of  the  slots  in  a  poly-phase  motor.  The 
broken  lines  mark  the  leakage  flux  threading  each  slot.  Now, 
let  us  assume  that  we  have  a  motor  with  48  slots  in  the  field, 
which  we  want  to  wind  as  a  two-pole,  four-pole,  or  eight-pole 
motor,  the  armature  being  provided  with  a  squirrel-cage  winding, 
thus  being  suitable  for  any  number  of  poles. 

If  the  number  of  ampere-conductors  per  slot  remains  the  same 
for  any  number  of  poles,  the  leakage  flux  per  slot  also  remains 
constant. 

The  total  number  of  ampere-turns  spread  over  the  circumfer- 
ence of  the  field  is  then  constant  whether  the  motor  has  two, 
four,  or  eight  poles. 

But  the  number  of  ampere-turns  per  pole  is  inversely  propor- 
tional to  the  number  of  poles;  hence,  the  induction  produced  by 
these  ampere-turns  in  the  air-gap  is  also  inversely  proportional 
to  the  number  of  poles.  In  other  words,  the  magnetic  field 
per  pole,  being  proportional  to  the  product  of  the  induction  in 
the  air-gap  into  the  pole-pitch,  varies  inversely  with  the  square 
of  the  pole-pitch. 

The  leakage  field,  as  we  have  seen,  is  constant  for  each  slot. 
Hence,  the  total  amount  of  leakage — the  sum  of  all  the  leakage 
fields  pertaining  to  each  slot — is  also  constant.  The  number  of 
leakage  lines  per  pole  is  proportional  to  the  number  of  slots  per 
pole. 

The  ratio  of  leakage  field  -=-  main  field  is  therefore  inversely 
proportional  to  the  pole-pitch  for  the  same  number  of  ampere- 
turns  per  slot. 

This  result  is  verified  by  the  following  series  of  tests: 

A. — Three-phase  current  motor  for  36  hp.,  380  volts  between 
the  lines,  six  poles,  42  ^>. 

Air-gap  A      =    0.62  mm. 
Pole-pitch  t  =  30.50  cm. 


100 


INDUCTION  MOTOR 


Volts  between 
the  lines 

Amperes,  field 

Amperes,  arma- 
ture 

Frequency 

81 
120 
150 
170 

36.0 
74.0 
106.0 
135.0 

95 
180 
260 
320 

42.0 
41.0 

383 

8.5 

0 

43.2 

If  we  draw  the  short-circuit  curve,  we  can  interpolate  for  380 
volts  the  short-circuit  current  of  380  amp.  We  thus  get  for  the 
leakage  factor  the  value 


B.  The  same  magnetic  frame  was  wound  for  24  hp.  190  volts 
between  the  lines,  10  poles,  50  ^. 

Air-gap  A  =  1.1  mm. 
Pole-pitch  t  =  18.3  cm. 


Volts  between  the  lines 

Amperes,  field 

Frequency 

20.0 

20.0 

51 

25.0 

31.5 

33.0 

50.5 

43.5 

75.0 

51 

66.0 

139.0 

83.0 

185.0 

95.0 

220.0 

Magnetizing  current:  31.2  amp.  at   190  volts.     The  short- 
circuit   current   at   190   volts   amounts  to  470   amp.     Hence, 

31.2 


470 


0.0664 


The  following  table  shows  the  results  of  the  tests: 


Air-gap  cm. 

Pole-pitch  cm. 

(7 

0.062 
0.110 

30.5 
18.3 

0.0224 
0.0664 

SHORT-CIRCUIT  CURRENT  AND  LEAKAGE  FACTOR     101 

For  equal  air-gaps  we  have 

<T!  =    0.0224  •  ^i  =  0.0396  (78) 


<7U  =  0.0664 
Hence, 

„„  _  0.0664 


00396        >' 

or,  in  other  words, 

^  =  ^  (80) 

0"i          hi 

The  leakage  factor  is  inversely  proportional  to  the  pole-pitch, 
or  directly  proportional  to  the  number  of  poles. 

By  the  above  experiments  it  has  been  demonstrated  that  the 
leakage  factor  is  directly  proportional  to  the  air-gap,  and  in- 
versely proportional  to  the  pole-pitch.  We  may,  therefore,  write 
the  formula  for  the  leakage  factor, 

<r  =  C-y  (81) 

in  which  equation  C  is  a  factor  dependent  upon  the  shape  and  size 
of  the  slots,  and  upon  a  great  many  other  conditions  of  which  we 
are  still  rather  ignorant.  For  practical  purposes,  however,  C 
can  be  determined  with  satisfactory  accuracy,  though  it  will  still 
be  left  to  the  designer  to  estimate  the  value  of  C  between  certain 
limits.  For  slots,  as  shown  in  Fig.  75,  C  varies  between  10  and  15. 
Since  this  formula  (81)  was  first  developed  by  the  author  in 
1896  and  1897,  it  has  been  constantly  tested  and  it  has'beftfx 
the  subject  of  considerable  discussion.  Other  formulas,  also 
almost  altogether  empirical,  have  been  suggested  by  Mr.  H.,:M. 
Hobart,  Dr.  Behn-Eschenburg,  and  others. 

H.  THE  DIFFERENT  LEAKAGE  PATHS 

There  are  at  least  three  pronounced  leakage  paths  in  an  in- 
duction motor. 

A.  The  leakage  of  the  end  windings.  The  reluctivity  of  this 
leakage  field  is  probably  approximately  proportional  to  the  length 
of  the  projecting  windings,  i.e.,  proportional  to  the  pole-pitch. 
At  least  it  is  a  fair  guess  that  the  longer  the  pole-pitch  the  greater 
is  this  flux.  However,  the  length  of  the  leakage  path  may  also 
increase  with  the  pole-pitch,  so  that  after  all  proportionality 
may  not  exist.  To  approach  the  subject  mathematically  requires 
the-  equipment  of  a  Maxwell  or  a  Heaviside. 


102 


INDUCTION  MOTOR 


B.  The  so-called  zig-zag  leakage,  by  which  is  probably  meant 
the  peculiarity  that  in  different  positions  of  the  rotor  some  of  the 
primary  (or  secondary)  flux  does  not  reach  around  the  secondary 
(or  primary)  conductors.  Whether  this  should  be  called  leakage 
at  all  is  problematical. 

C.  The  so-called  slot  leakage.  This  leakage  is  supposed  to 
close  around  the  slots  without  embracing  the  induced  member. 
To  attempt  to  obtain  a  physical  picture  of  this  leakage  is  also 
very  difficult.  The  leakage  lines  are  frequently  shown  by  writers 
to  intersect  the  main  field  which  is  an  assumption  contrary  to 


Conventional  and  incor- 
rect picture  of  leakage  flux 
diagram. 

FIG.  76. 


Correct  picture  of  leakage 
flux  diagram. 


our  conception  of  Faraday's  lines  or  tubes  of  force.  The  leakage 
lines  are  also  often  shown  as  indicated  in  Fig.  76,  which  is 
physically  an  error,  as  there  is  no  m.m.f.  for  the  closed  magnetic 
circuit  in  which  the  lines  are  shown  as  though  they  surrounded 
an  m.m.f. 

The  entire  subject  seems  somewhat  vague  and  problematical, 
greatly  confused  by  elaborate  speculations  based  upon  flimsy 
and  insufficient  premises  contradictory  of  our  fundamental 
conceptions  of  the  magnetic  field. 

It  is  possible,  however,  to  obtain  a  somewhat  vague,  physically 


SHORT-CIRCUIT  CURRENT  AND  LEAKAGE  FACTOR     103 

plausible,  picture,  at  least  in  rough  outline,  which  luckily  har- 
monizes sufficiently  well  with  observed  data. 
We  have  seen  that 

—  1  (14) 


*i»i  =  -  +  -  (82) 

P         Pi 

*„,_*!  +  *!  (83) 

P  P2 

where  $>i  and  $2  are  the  fictitious  primary  and  secondary  fluxes, 
Xi  and  Xz  the  primary  and  secondary  m.m.fs.,  p  the  reluctance 
of  the  common  magnetic  circuit,  pi  and  p2  the  reluctances  of  the 
primary  and  secondary  leakage  circuits. 

From  (82)  and  (83) 


i 

~  ~ 

But 

=  P  (84) 


Therefore 


i  =  1+  -  (86) 

Pi 

t  =  1  +  p- 

P2 

p~  )(1  +  p-)  -  1  (87) 

Pi  P2 


=  5-  +  P-  +  -5-  (88) 

Pi  P2  PlP2 

=  p(  --  1  —  )  approximately  (89) 

\Pl  P2/ 

Now, 


(90) 


where  b  is  the  width  of  core  of  the  motor. 


—7  =  Ki-t  for  the  end  connections  (91) 

—  =  Kr   --6  for  the  slots  (92) 


104 


INDUCTION  MOTOR 


where  d  is  depth  of  slot,  and  w  width  of  slot.     Hence,  adding 

-  --  K  t  +  K  -  b 
p  ^  w 

A  d    A 

r  ~.  04) 


(93) 


(95) 
(96) 


where  C  is  a  factor  which  varies  in  a  number  of  ways  depending 
on  features  of  design.  It  can  be  guessed  at  successfully  by  the 
experienced  designer,  but  it  is  not  amenable  to  sound  scientific 
calculation.  It  varies  between  the  limits  of  6  and  15. 

I.  FURTHER  EXPERIMENTAL  DATA 

From  an  exhaustive  investigation,  whose  results  are  given  in 
the  table  below  and  plotted  in  Fig.  77  the  following  formula  has 
been  derived: 


o.io 


0.05 


1 

2o"Diam.;  8  Poles;  12  Slott;  Size  .s"x  1.26* 
14  'Diam.;  6  Poles  ;  12     s"x  1.26" 

\ 

t-1 

A  =. 

,•: 

\ 

\ 

\ 

^  

*mi 

NT+T! 

0  5  10  15  20  25     in.         30 

Width  of  Iron 

FIG.  77. — The  leakage  factor  a  and  its  dependence  upon  the  width  of  the  motor. 

,  =  ^  (5.1^  +  5.65)  (97) 

in  which  A,  t,  and  6  are  to  be  substituted  in  inches. 


SHORT-CIRCUIT  CURRENT  AND  LEAKAGE  FACTOR     105 


60-Cycle  Induction  Motors 

Stators  12  slots  per  pole 

Slot  0.3  in.  by  1.125  in' 

t  =  7 . 8  in. 

A   =  0 . 03  on  one  side 


b  Inches 


2.00 
3.75 
3.75 
5.00 
6.50 
7.00 
7.50 
19.50 


0.0900 
0.0640 
0.0540 
0.0560 
0.0465 
0.0430 
0.0413 
0.0300 


It  is  well  to  keep  in  mind  that  a  is  approximately  equal  to  the 
magnetic  reluctance  of  the  common  magnetic  field  divided  by  the 
reluctance  of  the  parallel  leakage  fields 


Po 

p  = 


1  4-1 

Pi  P2 

1 

Pi  P2 

Po 


(98) 
(99) 

(100) 


K.  WINDING  THE  SAME  MOTOR  FOR  DIFFERENT  SPEEDS 

Formula  (96)  permits  us  to  determine  the  change  in  the  output, 
power  factor,  and  so  forth,  of  a  motor  wound  for  a  different 
number  of  poles,  for  instance,  for  eight,  four,  or  two  poles.  If 
the  field  has  48  or  72  slots,  it  can  easily  be  wound  so  as  to  satisfy 
this  demand.  We  will  assume  the  induction  in  the  air-gap,  or, 
which  is  the  same,  in  the  teeth,  to  remain  constant  for  all  three 
cases.  We  will  further  assume  that  the  motors  are  to  be  wound 
for  the  same  voltage.  Then  it  is  clear,  according  to  equation  (49), 
that  the  total  number  of  active  conductors  must  be  proportional 
to  the  number  of  poles;  in  other  words,  if  the  eight-pole  motor 
has,  for  instance,  720  conductors,  or  10  conductors  in  each  of  72 


106 


INDUCTION  MOTOR 


slots,  then  the  four-pole  motor  must  have  360,  and  the  two-pole 
motor  180,  in  order  to  get  the  same  induction  in  the  air-gap.  To 
calculate  the  relative  value  of  the  magnetizing  current  we  need 
only  know  the  number  of  active  conductors  n  per  pole,  see  equa- 

720 
tion  (45).     We  have  for  n  in  the  eight-pole  motor--  -  =90;  in 

o 

360 
the   four-pole    motor  -j-  =  90;    and    in   the    two-pole   motor 

180 

-£-  =  90.     Hence,  as  B  and  n  are  the  same  in  each  of  the  three 

cases,  it  follows  that  the  magnetizing  current  also  remains  the  same. 


Two  Poles 


FIG.  78. — The  primary  current  locus  of  the  induction  motor     The  leakage  factor 
and  the  circle  diagram  for  different  numbers  of  poles  on  the  same  frame. 

As  the  shape  and  size  of  the  slots  are  the  same  in  all  three  cases, 
the  factor  in  equation  (96)  for  the  leakage  coefficient  also  remains 
the  same.  Hence,  as  the  leakage  factor  is  proportional  to  the 
quotient  of  the  air-gap  divided  by  the  pole-pitch,  we  find  the 
short-circuit  current  to  be  inversely  proportional  to  the  number 
of  poles.  This  is  graphically  represented  in  Fig.  78.  A  glance 
at  the  diagram  teaches  us  that  the  maximum  energy  that  can 
be  impressed  upon  the  motor,  and,  therefore,  also  very  nearly 
the  output,  vary  in  proportion  to  the  pole-pitch.  According 
to  the  diagram  we  find  the  leakage  factor  for  the  two-pole  motor 

8  8 

equal  to  j™  =  0.05;  for  the  four-pole  motor  equal  to  ~~  =  0.10; 

g 
and  for  the  eight-pole  motor  equal  to  ^  =  0.20.     The  maximum 

power  factor  in  each  case  can  now  be  calculated  with  the  help 
of  formula  (19).  This  is  done  in  the  following  table: 


SHORT-CIRCUIT  CURRENT  AND  LEAKAGE  FACTOR     107 


Number  of 

Leakage 

Maximum 

Relative 

Revolutions 

poles 

factor 

power  factor 

output 

per  minute 

2 

0.05 

0.910 

80 

3,000 

4 

0.10 

0.835 

40 

1,500 

8 

0.20 

0.715 

20 

750 

The  output  is  therefore  proportional  to  the  number  of  r.p.m. 

It  may  not  be  amiss  here  to  remark  that  if  the  motor  is  ordi- 
narily wound  for  four  poles,  the  induction  in  the  iron  above  the 
slots  may  for  the  same  frame  become  too  high  in  the  two-pole 
motor,  thus  increasing  the  magnetizing  current,  and  possibly 
creating  undue  heating. 

L.  DRAWBACKS  OF  A  HIGH  FREQUENCY 

If  the  circumferential  speed  of  the  armature  is  limited,  and  this 
is  generally  the  case,  then  the  pole-pitch  is  also  limited  for  a 
given  number  of  r.p.m.  The  air-gap  cannot  indefinitely  be 
diminished,  hence,  a  high  frequency  necessitates  a  large  leakage 
factor  according  to  formula  (96). 1  We  labor  here  under  the 
same  difficulties  that  we  have  met  with  in  the  design  of  alter- 
nators for  high  frequencies.  It  is  doubtless  possible  to  build 
motors  for  frequencies  between  60  and  100,  but  the  higher  the 
frequency  the  lower  will  be  the  power  factor,  and  the  larger  will  be 
the  lagging  currents.  It  has  also  to  be  borne  in  mind  that 
motors  for  high  frequencies,  if  they  are  to  be  as  good  as  those  for 
low  frequencies,  must  be  made  not  inconsiderably  larger. 

Allowing  again  the  induction  in  the  air-gap  to  be  the  same 
for  different  frequencies,  which  is  a  more  or  less  challengeable 
proposition,  it  follows  from  formula  (49)  that  the  total  number 
of  active  conductors  around  the  circumference  of  the  field  must 
also  be  the  same,  for  the  pole-pitch  is  inversely  proportional  to 
the  frequency,  hence,  the  product  of  the  frequency  into  the 
number  of  lines  of  induction  per  pole  remains  the  same  if  the 
induction  in  the  air-gap  is  the  same. 

The  magnetizing  current,  however,  being  proportional  to  the 
ratio  of  the  induction  B  divided  by  the  number  of  active 

1  This  is  the  reason  why  Mr.  TESLA  and  the  Westinghouse  Company 
failed  to  design  a  successful  motor  between  1888  and  1890  as  a  frequency 
of  135  was  then  commonly  used.  See  B.  G.  LAMME,  "The  Story  of  the 
Induction  Motor,"  A.  L  E.  E.  Journal,  March,  1921. 


108 


INDUCTION  MOTOR 


conductors  per  pole,  is  thus  inversely  proportional  to  the  fre- 
quency. The  leakage  factor  is,  according  to  formula  (96), 
directly  proportional  to  the  pole-pitch,  or  inversely  propor- 
tional to  the  frequency — because  the  pole-pitch  is,  in  the  case 
under  consideration,  inversely  proportional  to  the  frequency — 
hence,  it  follows  that,  as  the  magnetizing  current  has  been 
shown  to  be  proportional  to  the  frequency,  the  diameter  of  the 
semi-circle  remains  constant  for  all  frequencies. 

Figure  79  shows  the  polar  diagram  for  the  same  motor,  but 
for  different  frequencies.  The  maximum  energy  that  the  motor 
is  capable  of  taking  in,  and,  therefore,  also  the  maximum  output, 
is  the  same  for  100^,  50^,  or  25^.  But  the  maximum  power 
factor  is  considerably  smaller  for  the  high  frequencies,  as  a 


-1GOA- 


FIG.  79. — The   leakage   factor   of   the   induction   motor.     The    circle   diagram 
for  different  frequencies. 

glance  at  the  diagram  shows.     The  following  table  shows  the 
leakage  factor  and  the  power  factor  in  relation  to  the  frequency : 


Frequency 

Leakage  factor 

Maximum  power  factor 

25 
50 
100 

0.05 
0.10 
0.20 

0.910 
0.830 
0.715 

I  wish  to  call  attention  to  the  fact  that  the  motor  for  the 
higher  frequencies  is  here  represented  less  unfavorably  than  it 
really  is,  because  the  induction  in  the  air-gap  has  to  be  reduced 
if  the  motor  is  to  be  wound  for  a  higher  frequency.  The  im- 
mense lagging  currents  invariably  bound  up  with  the  higher 
frequency  are  very  clearly  shown  in  the  diagram. 


SHORT-CIRCUIT  CURRENT  AND  LEAKAGE  FACTOR     109 

It  is  to  be  remembered  that  the  current  in  the  armature  is 
dependent  upon  the  leakage  factor,  since  the  transformation 
factor  vi  forms  part  of  the  leakage  factor.  (See  Fig.  16.)  The 
transformation  factors  v\  and  v2  are  connected  with  a  through 
equation  (14), 

—  1  (14) 


Hence,  it  follows,  as  z'2=  A  D  '  ~9\t  see  Fig.  16,  that  the  current 

in  the  armature  is  larger  for  the  motor  running  at  a  high  fre- 
quency than  for  that  running  at  only  25^.  In  our  case,  setting 
vi  =  v2,  we  get  for  v  at  25~,  0.978,  and  at  100~,  0.912,  therefore 
the  current  in  the  armature  of  the  motor  for  100  ~  is,  for  the 
same  AD,  1.07  times  larger  than  for  25^.  This  corresponds  to 
an  increased  armature  loss  of  about  14  per  cent.  But  as  the 
primary  current  is  also  larger  for  100^  than  for  25^,  the  arma- 
ture loss  is  still  greater  than  here  calculated.  Thus  to  the  draw- 
back of  large  lagging  currents,  there  has  to  be  added  the  further 
drawback  of  considerably  larger  losses. 

The  foregoing  experiments  and  considerations  are,  within 
my  knowledge,  the  first  attempt  to  deal  in  a  rational,  systematic 
manner  with  the  conditions  underlying  the  leakage  in  poly- 
phase motors.  I  am  far  from  claiming  for  this  treatment  com- 
pleteness of  conclusiveness;  on  the  contrary,  I  deem  it  a  necessity 
to  revise  it  by  the  light  of  forthcoming  experience.  I  am  toler- 
ably confident  that  the  main  propositions  will  be  proved  true, 
while  minor  points  may  need  some  qualification. 

Considering  the  immense  complexity  of  the  phenomena  in 
poly-phase  motors,  the  greater  or  less  arbitrariness  which  hangs 
about  most  of  our  assumptions  which  have  to  be  made  in  order 
to  be  able  to  calculate  at  all,  I  cannot  forbear  from  wondering 
that  so  approximate  a  solution  can  be  attained  at  all.  It  may  be 
that  there  are  errors  inherent  in  our  fundamental  assumptions 
which  all  so  counteract  one  another  as  to  cause  the  result  of 
calculation  to  deviate  but  little  from  experiment  and  observa- 
tion. This  view  will  commend  itself  to  those  who  are  familiar 
with  some  branch  of  physiology,  for  instance,  physiological  optics; 
here  we  have  the  testimony  of  Helmholtz  that  the  eye,  having 
"every  possible  defect  that  can  be  found  in  an  optical  instru- 
ment," yet  gives  us  a  fairly  accurate  image  of  the  outer  world 
because  these  various  defects  balance  one  another  almost 
completely. 


110  INDUCTION  MOTOR 

The  above  remarks  will  be  distasteful  to  those  who  have 
accustomed  themselves  to  look  upon  only  one  side  of  a  question, 
and  who  try  to  shut  their  eyes  to  the  inevitable  uncertainties 
that  beset  us  in  all  intellectual  problems.  I  was  once  taken  to 
task  by  a  critic  for  having  adduced  experimental  evidence  qualify- 
ing my  theory,  and  narrowing  the  limits  of  its  application,  and  I 
was  told  that  these  experiments  invalidated  my  argument, 
while  my  intention  —  to  lay  stress  upon  the  incompleteness  and 
the  shortcomings  of  the  theory  —  was  obviously  not  even  thought 
of  by  my  critic.  Politicians  and  propagandists  may  have  to 
hide  the  weak  sides  and  spots  of  their  arguments,  but  men  of 
science  are  bound  to  point  them  out  and  to  expose  them. 

M,  HISTORICAL  AND   CRITICAL  DISCUSSION  OF  THE  LEAKAGE 

FACTOR 

In  The  Electrical  Engineer,  London,  Dec.  11,  1903,  Mr.  H.  M. 
Hobart  discussed  the  equation  (96)  in  his  usual  thoughtful 
manner.  He  points  out  that  too  much  weight  has  been  placed 
upon  the  pole-pitch  and  that  the  formula  might  lead  to  motors 
of  too  large  a  diameter.  He  says  :  '  (  In  the  following  article  the 
writer  wishes,  in  the  first  place,  to  emphasize  the  importance  of 
that  part  of  the  total  inductance  which  is  due  to  the  end  con- 
nections, and,  in  the  second  place,  to  develop  a  simple  and  practi- 
cal method  by  which  the  best  dimensions  may  be  decided.  On 
p.  36  of  Behrend's  excellent  treatise  on  induction  motors,  the 
following  formula  for  calculating  a  is  given  : 


where  C  is  a  figure  which  is  dependent  on  the  slot  dimensions 
and  other  conditions,  A  is  the  length  of  air-gap,  and  t  the  pole- 
pitch.  Behrend  estimates  that  C  varies  between  10  and  15 
for  half  -closed  slots.  This  formula  is  extremely  useful  on  account 
of  its  simplicity,  especially  if  experimental  results  are  available 
by  which  to  decide  the  value  of  C." 

Mr.  Hobart  then  proceeds  to  give  in  a  table  the  ratio  of  - 

t 

and  determines  C  accordingly.  He  closes  with  the  statement: 
"It  is,  however,  preferable  to  keep  the  formula  as  simple  as 
possible,  so  that  the  writer  thinks  it  better  to  retain  Behrend's 
original  formula,  together  with  the  values  of  C  got  from  Table 
I  and  Fig.  1." 


SHORT-CIRCUIT  CURRENT  AND  LEAKAGE  FACTOR     111 


The  Institution  of  Electrical  Engineers,  London,  held  a  meeting 
Jan.  14,  1904,  at  which  Prof.  Silvanus  P.  Thompson  presented  a 
paper  prepared  by  Dr.  H.  Behn-Eschenburg,  "On  the  Magnetic 
Dispersion  in  Induction  Motors,  and  its  Influence  on  the  Design 
of  these  Machines."  Dr.  Behn-Eschenburg  starts  with  a  slightly 
different  definition  of  the  coefficient  of  leakage  from  that  used 
by  us. 

<TQ  =  1 fV-^  (101) 

LiiLz 

(102) 


.  (To    = 


(103) 


FIG.  80. 
FIGS.  80  and  81. — Dispersion  of  the  flux. 


FIG.  81. 
(After  Behn-Eschenburg) . 


This  <TO  is  the  ratio  of  the  magnetizing  current  to  the  short-circuit 
current  while  our  d  is  the  ratio  of  the  magnetizing  current  to  the 
diameter  of  the  circle.  The  following  six  figures  which  are 
instructive  are  taken  from  the  Behn-Eschenburg  paper.  The 
investigation  results  in  the  formula 


cro  =  —^ 


6A 
b 


(104) 


in  which  n  is  the  mean  number  of  slots  for  stator  and  rotor  in  the 
pole-pitch;  £  the  opening  in  centimeters  of  the  half -closed  slot. 
The  remaining  notation  is  identical  with  ours. 

A  comparison  of  Dr.  Behn-Eschenburg's  formula  shows  that 
it  is  practically  identical  with  ours  equation  (96). 

In  the  discussion  Mr.  H.  M.  Hobart  said: 

"The  data  contributed  in  Dr.  Behn-Eschenburg's  paper  are  of  great 
value  to  those  engaged  in  induction  motor  design.  The  exhaustive 


112 


INDUCTION  MOTOR 


series  of  comparative  tests  which  are  described  in  the  paper,  throw  more 
light  on  the  subject  than  any  contributions  since  the  publication  of 
Behrend's  investigations.  Dr.  Thompson  has  mentioned  Behrend's 
formula  for  the  determination  of  a.  This  formula  of  Behrend's  is  in 
refreshing  contrast  with  many  that  have  been  proposed,  and  it  is  satis- 
factory to  note  that  Dr.  Behn-Eschenburg's  formula  is  also  of  fairly 
moderate  length."  Mr.  Hobart  then  proceeds  to  recommend  his  for- 


FIG.  82. 


FIG.  83. 
FIGS.  82  and  83. — Dispersion  of  the  flux.      (After  Behn-Eschenburg). 

mula  already  referred  to.  "It  appears  fairly  certain  that  the  use  of 
Behrend's  formula  in  its  present  form,  namely: 

(7  =  CC1  y  (105) 

the  constants  C  and  C1  being  determined  respectively  from  the  curves 
of  Figs.  1  and  2,  will,  for  practically  all  commercial  types  of  induction 
motors  yield  very  good  results." 

Prof.  Silvanus  P.  Thompson  stated  that,  "What  has  been  done  by 
Kolben,  Behrend,  and  others  has  been  to  state  certain  very  simple 
empirical  rules,  of  which  Behrend's  is  perhaps  the  best." 

This  formula  for  a  was  developed  and  used  by  me  in  1896  and 
1897  in  the  testing  department  of  the  Oerlikon  Company  on  the 


//v- 


(Facing  page  112) 


SHORT-CIRCUIT  CURRENT  AND  LEAKAGE  FACTOR     113 

basis  of  my  circle  diagram,  with  which  engineers  were  then  quite 
unfamiliar.  It  was  published  by  me  in  January,  1900,  in  lectures 
delivered  at  the  University  of  Wisconsin,  republished  in  the  Elec- 


FIG.  84. 


FIG.  85. 
FIGS.  84  and  85.  —  Dispersion  of  the  flux.      (After  Behn-Eschenburg). 


trical  World,  and  afterwards  in  book  form.  Dr.  Behn-Eschen- 
burg published  a  slightly  different  formula  in  the  pamphlet  Sur 
le  Calcul  des  Machines  electriques,  Oerlikon,  June,  1900. 


114  INDUCTION  MOTOR 

N.  BIBLIOGRAPHY 

R.   E.    HELLMUND:   Reference    should    be    made    also    to  the  papers  of. 

"Graphical   Treatment  of  the  Rotating  Field,"  Trans.  Am.  Inst.  El. 

Engrs.,  June-July,  1908. 

"Graphical  Treatment  of  the  Zigzag  and  Slot  Leakage  in  Induction 

Motors,"  London,  Institution  of  Electrical  Engineers,  June,  1909  and 

May,  1910. 
PROF.    C.    A.    ADAMS:  "The   Leakage   Reactance   of  Induction  Motors," 

International  Electrical  Congress,  St.  Louis,  September,  1904. 

"A  Study  in  the  Design  of  Induction  Motors,"  Trans.  Am.  Inst.  El. 

Fngrs.,  June,  1905. 
R.   GOLDSCHMIDT:  "The  Leakage  of  Induction   Motors,"  London:  "The 

Electrician,"  Printing  &  Pub.  Co.,  Ltd.     No  date. 
DR.    A.    S.    MCALLISTER:  "Alternating    Current    Motors,"    New    York: 

McGraw-Hill  Book  Co.,  Inc.,  1909. 
E.  ARNOLD  ET  J.-L.  LA  COUR:  "Les  Machines  d'Induction,"  Paris,  Librairie 

Ch.  Delagrave,  1912. 
ALEXANDER  GRAY:  "Electrical  Machine  Design,"  New  York:  McGraw-Hill 

Book  Co.,  1913. 


CHAPTER  VIII 
THE    DOUBLE    SQUIRREL-CAGE    INDUCTION    MOTOR 

An  ingenious  suggestion  was  made  in  the  early  nineties  by 
M.  Dolivo-Dobrowolsky,1  which  has  recently  found  an  important 
application  in  the  quarter-phase  induction  motors  propelling  the 
battleship  U.  S.  S.  New  Mexico.2  It  consists  in  the  use  of  two 
squirrel-cage  windings  on  the  same  rotor.  The  winding  near  the 
surface  has  a  high  resistance  and  high  leakage  reluctance,  while 
another  winding  placed  beneath  this  winding  has  a  low  resistance 
and  a  low  leakage  reluctance.  The  high  resistance  of  the  outer 
winding  is  obtained  by  the  use  of  an  alloy  of  approximately  18 
per  cent  german  silver,  while  the  low  leakage  reluctance  of  the 
second  winding  is  obtained  by  a  separate  leakage  path. 

The  ostensible  idea  of  the  arrangement  is  to  utilize  the  outer 
high  resistance  winding  in  starting,  and  the  inner  low  resistance 
winding  for  running.  It  appears  plausible  that,  in  view  of  the 
high  frequency  of  the  secondary  currents  when  starting,  and  the 
low  secondary  frequency  when  running,  these  conditions  may  be 
satisfactorily  fulfilled.  This  condition  can  be  determined  only 
by  a  careful  quantitative  analysis  of  the  arrangement.  Suffice 
it  to  say  that,  in  spite  of  the  fact  that  the  arrangement  has.  been 
known  for  over  25  years,  it  has  found  no  application  until  recently. 

Figure  86  represents  the  arrangement  of  the  windings  in 
the  slots,  and  Fig.  87  represents  the  distribution  of  the  magnetic 
flux  in  the  different  circuits.  It  is  clear  that  the  Low  Resistance 
Winding  III,  whose  m.m.f.  is  Xs,  is  closed  through  the  leakage 
path  p3,  whose  leakage  flux  is  /3.  The  flux  produced  by  X8,  if 
acting  alone,  through  the  reluctance  p  is  <£3,  not  shown  in  the 
figure.  The  m.m.f.  Xz  of  the  High  Resistance  Winding  II  acts 
in  series  with  the  m.m.f.  X3  so  that  its  flux  3>2,  not  shown  in  the 
figure  would,  be  $2  =  X%  -r-  p,  if  acting  alone,  and  therefore  if 
only  X2  and  X3  were  acting  upon  the  circuits,  the  flux  <£2  +  $3 
=  (Xz  +  Xs)  -r-  p,  would  exist  in  the  air-gap.  These  fluxes  are 
prevented  from  becoming  established  through  the  existence  of 
the  m.m.f.  Xi. 

1  U.  S.  Patent  No.  427,978,  May  13,  1890. 

2  "General   Characteristics   of   Electric   Ship    Propulsion    Equipments." 
By  E.  F.  W.  ALEXANDERSON.     General  Electric  Review,  April,  1919. 

115 


116 


INDUCTION  MOTOR 


To  fix  ideas  we  also  represent  in  Fig.  88  the  electric  circuits 
to  which  this  type  of  motor  is  equivalent,  assuming  a  ratio  of 
transformation  of  one  to  one.  As  both  rotor  windings  have 


Stator  Slots 
-t— Winding  I 


High  Resistance 
Winding  II 

Rotor  Slots 


w  Resistance 
Winding  III 


FIG.  86. — Arrangement  of  slots  of  double-squirrel  cage  motor. 

the  same  slip,  the  variation  in  speed  corresponds  to  a  trans- 
former with  variable  resistances,  as  indicated  in  the  Fig.  88,  viz., 
r2  -5-  s  and  n  +  s  (106) 


^AAAAA/WW^/W 


FIG.  87. — The  leakage  paths  of  the  magnetic  circuit  of  the  double-squirrel  cage 

motor. 

We  begin  with  Winding  III.     Its  resultant  flux  is  F3.     This 
flux  sets  up  the  e.m.f.  which  sends  current  through  the  ohmic 


DOUBLE  SQUIRREL-CAGE  INDUCTION  MOTOR          117 

resistance  of  the  Low  Resistance  Winding  III.     This  e.m.f.  is 
equal  to 

e3  =  2.12(~i  -  ~2)z3F310-8  volts  (107) 


FIG.  88. — Equivalent   transformer   circuits   for   double-squirrel   cage   induction 

motor. 


FIG.  89. — The  diagram  of  fluxes  of  the  double-squirrel  cage  motor. 

The  current  produced  by  this  e.m.f.  is  equal  to  i3  =  e3  -r-  r3. 
The  leakage  field  fs  through  the  path  of  reluctance  p3  is  in  time- 


118  INDUCTION  MOTOR 

phase  with  and  proportional  to  i3  and  it  is  to  be  estimated  in  the 
usual  manner.  In  Fig.  89  it  is  represented  by  ab  =  /3.  OB 
in  Fig.  89  is  the  flux  F2,  as  also  indicated  in  Fig.  87. 

If  X*  created  by  is  were  acting  alone,  it  would  circulate  a  flux 
<i>3,  Fig.  89.     Likewise,  the  e.m.f.  e%  induced  by  F2, 

e2  =  2.12(~i  -  ~2)22/MO-8  volts  (108) 
produces  a  current  i2  =  e2  -£•  r2,  and  an  m.m.f.  X2  which,  acting 
alone,  would  circulate  a  flux  <i>2  in  quadrature  with  F2,  as  shown 
in  Fig.  89.  Combining  <£3  and  $2  vectorially,  gives  f>2  +  $3, 
represented  in  the  diagram  in  line  with  be. 

As  in  the  general  flux  theory  of  the  induction  motor,  so  here, 

=  $3  +  /3  (109) 

=  /3  (110) 

=  (f2  +  *8)  +  /2              (HI) 

=  /2  (112) 

/3  =  ab  (113) 

/2  =  be  (114) 

/i  =  ce  (115) 

As  before  in  the  theory  of  the  induction  motor,  it  follows  readily 

=  cd  (116) 

The  diagram  of  Fig.  89  shows  clearly  and  significantly  the 
composition  of  the  fluxes  Fs,  /3,  /2,  and  /i  into  the  primary  result- 
ant flux  FI  which  induces  the  counter  e.m.f.,  which  balances  the 
primary  impressed  e.m.f.  As  before,  the  primary  resistance  TI 
is  neglected.  It  can  easily  be  taken  into  account  as  in  Chap.  III. 
An  inspection  of  the  diagram  shows  the  influence  of  the  low 
reluctance  of  the  Leakage  Path  III.  To  show  the  effect  of  this 
leakage,  a  complete  performance  of  a  motor  has  been  worked  out 
for  a  range  of  slip  from  zero  to  infinity  for  given  motor  character- 
istics, as  follows: 

r3  =  0.06 

r2  =  0.6 

v3  =  1.3 

v2  =  1.1 

»i  =  1.1 

*.-^ 


7*2 

33.5 


DOUBLE  SQUIRREL-CAGE  INDUCTION  MOTOR          119 


These  characteristics  correspond  closely  to  a  large  slow  speed 
motor  with  the  exception  that  the  leakage  is  assumed  somewhat 
larger  than  it  would  be  in  reality,  as  well  as  the  reluctance  of  the 
main  magnetic  circuit.  The  real  motor,  therefore,  would  have 
a  higher  power  factor. 

The  following  table  is  obtained  from  corresponding  points 
carefully  worked  out  : 


Slip 

Cosine  $\ 

Prim,  cur- 
rent 

Torque 

oo 

0.0000 

0 

Current 

and    Cos   \l/i 

50.0 

0.0375 

192.0 

66 

do     not 

correspond 

10.0 

0.2250 

180.0 

400 

to  these 

points. 

4.0 

0.3300 

143.0 

470 

3.0 

0.3350 

129.0 

430 

2.0 

0.3150 

116.0 

370 

256 

1.5 

0.2950 

109.0 

312 

350 

1.0 

0.2500 

105.0 

263 

500 

132 

0.9 

0.2300 

104.0 

244 

546 

150 

0.8 

0.2370 

104.0 

244 

600 

165 

0.7 

0.2230 

102.0 

225 

643 

180 

0.6 

0.2350 

100.0 

232 

710 

205 

0.5 

0.2380 

98.5 

230 

760 

230 

0.4 

0.2400 

96.0 

235 

793 

280 

0.3 

0.2900 

91.0 

270 

776 

320 

0.2 

0.3700 

87.0 

320 

635 

345 

0.1 

0.4800 

69.0 

337 

370 

275 

0.06 

0.5200 

54.0 

276 

210 

190 

0.0 

0.0000 

33.5 

0 

0 

0 

These  results  are  represented  in  the  polar  diagram,  Fig.  90.  An 
analysis  of  this  figure  yields  the  following  results: 

First,  the  locus  of  the  primary  current  is  no  longer  a  circle. 

Secondly,  if  the  Low  Resistance  Winding  III  did  not  exist, 
the  locus  of  the  primary  current  would  be  the  circle  about  0'  as 
center,  with  a  diameter 


aft  = 

where  <r  =  v&z  —  1 

=  1.1  X  1.1  -  1 
=  0.21 


(117) 


This  circle  is  shown  in  the  figure. 


120  INDUCTION  MOTOR 

Thirdly,  if  the  High  Resistance  Winding  II  did  not  exist,  the 
locus  of  the  primary  current  would  be  the  circle  about  0"  as 
center,  with  a  diameter 

Oa 
ac  =  - 

<7 

where  a-  =  ViV^  —  1 

=  1.1  X  1.3  -  1 
=  0.43 

This  circle  is  also  shown  in  the  figure. 


S=oo 


FIG.  90. — The  polar  diagram  of  the  locus  of  the  primary  current  of  the  double 

squirrel  cage  motor. 

Fourthly,  it  appears  that  the  actual  performance  of  the  motor  is 
even  less  satisfactory  than  that  which  would  correspond  to  a 
motor  equipped  only  with  the  Low  Resistance  Winding  III,  as 
is  natural  enough,  as  there  exists  besides  the  leakage  path  II. 

Fifthly,  adding  the  two  leakage  paths  II  and  III  we  obtain 

v2  =  1.40 

ff  =  1.1  X  1.4  -  1 
=  0.54 

The  circle  corresponding  to  such  a  motor  is  shown  over  ad  with 
0'"  as  center. 

Sixthly,  the  torque  of  the  motor  in  synchronous  watts  for 
any  given  speed  or  slip  is  represented,  as  in  the  ordinary  induc- 
tion motor,  by  the  watt  component  of  the  primary  current.  An 
analysis  of  the  relations  between  torque  and  slip  shows  the  inter- 
esting fact  that,  though  the  maximum  torque  obtainable  from  a 
certain  frame  is  greatly  reduced  by  the  Low  Resistance  Winding 


DOUBLE  SQUIRREL-CAGE  INDUCTION  MOTOR          121 


III,  the  slip  of  the  motor-  while  running  near  synchronism  is 
considerably  less  than  it  would  have  been  had  a  single-rotor 
winding  of  high  enough  resistance  been  used  to  obtain  the  same 
starting  torque  with  the  same  starting  current  as  in  the  case  of 
the  double  squirrel-cage  motor. 

A  single  high  resistance  winding  in  the  position  of  winding  II 
would,  on  the  other  hand,  give  a  performance  circle  ab.  Its 
resistance  could  be  so  chosen  that  its  slip  at  normal  load  were 
equal  to  that  of  the  double  squirrel-cage  motor.  However,  in 
this  case,  the  starting  torque  and  the  starting  current  would 
be  approximately  twice  those  of  the  double  squirrel-cage  motor 


FIG.  91. — Upper    curve:  Torque    of    a    double-squirrel    cage    motor.     Lower 
Curve:  Torque  of  a  single-squirrel  cage  motor. 

and  it  would  be  necessary  to  reduce  the  voltage  40  per  cent  at 
starting  in  order  to  obtain  equivalent  conditions  for  the  single 
winding  motor  with  the  double  squirrel  cage.  As  this  requires 
the  use  of  auto-transformers,  it  may  be  less  desirable  under 
certain  conditions  than  the  double  squirrel-cage  arrangement. 

A  torque  curve  is  shown  in  Figs.  91  and  92  which  brings  out 
the  interesting  and  singular  fact  that  the  double  squirrel-cage 
motor  cannot  accelerate  its  starting  torque  as  the  dip  in  the 
torque  curve  shows  that  the  torque  at  60  per  cent  slip  is  13  per 
cent  less  than  at  100  per  cent  slip. 

For  regular  commercial  use  this  type  of  motor  would  seem  to 
be  unsuitable  in  view  of  the  great  reduction  of  maximum  output 
as  a  result  of  the  great  leakage  of  the  low-resistance  winding. 

For  special  conditions,  like  the  electric  propulsion  of  ships,  it 
may  have  the  advantage  claimed  for  it,  viz.,  the  elimination  of  a 


122 


INDUCTION  MOTOR 


large  control  rheostat,  but  in  a  sense  it  embodies  in  its  own 
frame  a  variable  reactance  which  takes  the  place  of  a  starting 
auto-transformer  installed  outside  the  motor.  Nor  is  it  alto- 
gether clear  whether  smaller  and  lighter  motor  frames  with  a 
single  high  resistance  winding  of  low  leakage  would  not  yield 
equally  good  results  in  the  individual  case  in  which  one  or  two 


FIG.  92. — Torque  and  current  of  double-squirrel  cage  induction  motor. 
1.  Lower  curve  is  the  current.  2.  Upper  curve  in  heavy  lines  the  torque  of  the 
double-squirrel  cage  motor.  3.  Dotted  curve  the  torque  of  the  same  motor 
without  the  inner  cage.  4.  Short  solid  curve  torque  of  the  same  motor  without 
the  outer  cage. 

motors  may  be  operated  from  one  generator  as  a  unit,  if  it  is 
desired  to  dispense  with  an  external  rheostat  in  the  rotor  circuit. 
To  sum  up,  the  performance  of  a  double  squirrel-cage  rotor 
as  to  power  factor  and  overload  torque  approaches  very  closely 
the  performance  of  an  induction  motor  with  a  single-rotor  wind- 
ing, the  constants  of  which  are  the  same  as  those  of  the  high- 
leakage  winding.  Thus  power  factor  and  overload  torque  are 


DOUBLE  SQUIRREL-CAGE  INDUCTION  MOTOR          123 

greatly  reduced,  in  comparison  with  a  normal  low-leakage  motor. 
The  advantage  consists  in  lowering  the  slip  near  synchronism, 
and  thus  raising  the  efficiency  of  the  motor,  on  the  assumption 
that  the  starting  torques  of  both  types  are  approximately  the 
same,  as  well  as  their  starting  currents.  While,  if  the  slip  is  the 
same  at  normal  load,  the  starting  current  of  the  double  squirrel- 
cage  motor  is  increased  over  that  of  the  low-resistance,  high- 
leakage  type  of  motor.  All  this  is  clearly  shown  in  our  diagrams. 


CHAPTER  IX 


POLY-PHASE   COMMUTATOR    MOTORS 
PROPERTIES  OF  COMMUTATORS 

A.  THE  ACTION  OF  THE  COMMUTATOR 

In  this  chapter  we  shall  study  the  characteristics  of  an  arma- 
ture wound  like  a  direct-current  machine,  rotating  in  a  stator 
without  windings.  The  commutator  connected  to  this  armature 
carries  a  set  of  poly-phase  brushes  stationary  in  space  and  poly- 
phase current  of  a  given  frequency  is  supplied  to  the  armature 
through  these  brushes. 

Neglecting  for  the  present  the  phenomena  of  commutation, 
we  see  at  a  glance  that  the  action  of  commutator  and  stationary 


I C 


FIG.  93. — Poly-phase    commutator    motor.     The    action    of    the    commutator. 

brushes  consists  in  creating  stationary  groups  of  coils,  Fig.  93, 
as  though  these  coils  or  groups  of  coils  belonged  to  the  stationary 
part  of  an  induction  motor.  Supposing  then  that  the  supply 
circuit  has  a  given  constant  frequency  ^i  and  that  the  commu- 
tator-rotor revolves  with  angular  velocity  oj2.  The  supply 
frequency  being  ^i,  a  rotating  field  of  angular  velocity  coi  = 
2ir~i  is  set  up  in  our  two-pole  model  and  there  is  a  slip  s  = 

— -  set  up  between  the  rotor  coils  and  the  rotating  magnetic 

coi 

field  F.  Between  brushes  A  —  B,  B  —  C,  and  C  —  A,  there  is 
therefore  going  to  be  induced  by  rotation  an  e.m.f.,  Eq.  (47). 

e  =  1.84(~! 2)Z2^10-8  volts  (118) 

124 


POLY-PHASE  COMMUTATOR  MOTORS 


125 


+e 


BELOW 
Synchronism 

tu 


where,   as   usual,  22  is  the  number  of  active  rotor  conductors 
between  A  and  B,  B  and  C,  and  C  and  A. 
Equation  (118)  may  also  be  written 

e  =  1.84~is-z,-F10-8  volts  (119) 

where  s  is  the  slip. 

If  in  Fig.  94  F  is  the  flux  of  the  rotating  field  it  is  obvious  that 
at  standstill,  neglecting  losses,  the  e.m.f . 
to  be  impressed  upon  the  rotor  through 
the  stationary  brushes  must  lead  by  a 
quarter-time  phase,  as  the  magnetizing 
current  i  must  be  a  watt-less  current 
and  therefore  lagging  by  a  quarter-time 
phase. 

However,  with  increasing  speed  of  the 
rotor  the  slip  approaches  zero  at  syn- 
chronism, while  above  synchronism  the 
phase  of  e  reverses.  Thus  we  are  led 
to  the  brilliant  discovery  of  Leblanc 
later  utilized  by  M.  Marius  Latour  and 
A.  Scherbius  that  such  a  commutator 
armature,  if  rotated  at  a  negative  slip  in 

its  own  field,  acts  like  a  condenser,  or  commutator  supplied  with 
expressing   it  more  rigorously,  it  takes 
lagging   currents  from  the  supply   cir- 
cuit or  it  generates  leading  currents  through  its  rotation  in  its 
own  field. 

The  reasoning  pursued  above  is  based  on  the  assumption  that 
the  magneto-motive  force  belts  of  the  winding  generate  solely 
a  rotating  field  and  that  there  are  no  single-phase  fields  remaining 
over  as  leakage  fields  whose  effect  is  prominent  at  constant 
frequency  instead  of  slip  frequency.  The  apparent  reactance  per 
phase  X2Q  of  the  rotor  in  regard  to  the  rotating  field  F  is  equal  to 


ABOVE 


synchronism 


fcr, 


(120) 

In  other  words,  X2°  is  proportional  to  the  slip  and  becomes 
negative  at  negative  slips. 

Supposing,  however,  that,  failing  to  obtain  a  perfect  rotating 
field,  there  remain  local  fields  entirely  independent  of  the  slip. 


126 


INDUCTION  MOTOR 


These  fields  would  set  up  an  e.m.f .  proportional  to  the  impressed 
frequency  and  current  and  therefore  the  reactance  X2"  corre- 
sponding to  the  effect  of  these  fields  is  constant. 


=  constant 


(121) 


Considering  now  the  effect  of  the  short-circuit  currents  under 
the  brush  as  produced  by  the  rotating  field  F,  they  effect  the 
generation  of  an  e.m.f.  proportional  to  the  product  of  the  slip 
and  the  field  divided  by  the  impedance  of  the  short-circuited 
coil.  Assuming  this  impedance  to  be  primarily  a  resistance,  it 
is  not  unreasonable  to  assume  it  to  be  constant.  Hence,  the 


+1 


S- 


-1 


FIG.  95. — The  local  leakage  reactance  of  the  poly-phase  commutator  rotor 
as  composed  of  its  various  parts  showing  the  effect  of  the  variable  and  constant 
portions.  (Leblanc-Latour) . 

effect  of  the  short-circuited  turns  under  the  brushes  corresponds 
to  a  reactance  XJ  which  is  proportional  to  the  slip, 


-XV  = 


(122) 


Therefore,  the  total  reactance  of  the  rotating  commutator  arma- 
ture is  equal  to 


2 
K2" 


(123) 


In  rectangular  coordinates,  these  relations  may  be  represented 
as  shown  in  Fig.  95,  from  which  it  appears  that  the  reactance 
existing  at  s  =  0,  i.e.,  at  synchronism,  is  equal  to  the  constant 
part  of  the  total  reactance  of  the  commutator  rotor. 


POLY-PHASE  COMMUTATOR  MOTORS 


127 


B.  PROPERTIES  OF  PHASE  LAG  OR  LEAD  OF  THE  POLY -PHASE 

COMMUTATOR 

The  use  to  which  Leblanc,  Latour,  Scherbius,  and  the  Brown, 
Boveri  Company  have  put  these  rotors  is  now  apparent.  Ope- 
rated by  a  small  motor  at  high  speed  with  a  large  negative  slip, 
either  with  a  mechanically-stationary  magnetic  circuit,  or 
with  conductors  embedded  in  the  revolving  iron  ring,1  they  are 
connected  in  series  with  the  rotor  winding  of  a  slip  ring  type  of 


FIG.  96. — The  effect  of  leading  currents  in  the  rotor  of  a  slip-ring  type  of  induc- 
tion motor  upon  the  primary  currents. 

induction  motor.  Thus  they  are  excited  by  low  frequency 
currents  and  an  e.m.f.  proportional  to  the  rotor  current  but  in 
lagging  quadrature  therewith  is  induced  in  the  rotor.  The 
polar  diagram  is  drawn  in  Fig.  96,  which  already  has  been 
discussed  in  Chap.  Ill,  C.  It  shows  the  locus  of  the  primary 
current  for  both  induction  motor  and  induction  generator  ranges 
so  long  as  the  current  which  flows  into  the  Leblanc  commutator- 
1  A.  SCHERBIUS,  "The  Electrician,"  London,  1912,  p.  582. 


128  INDUCTION  MOTOR 

rotor  leads  the  impressed  e.m.f.  at  its  brushes.  This  condition 
must  be  fulfilled  in  order  to  obtain  the  benefit  of  the  commutator 
action. 

Saturation  of  the  magnetic  circuit  of  the  Leblanc  rotor  has  been 
proposed  by  Scherbius  in  order  to  obtain  as  rapidly  as  possible  a 
high-power  factor  at  small  loads  without  low-leading  power 
factors  at  greater  loads,  see  Fig.  160,  which  is  self-explanatory. 

C.  COMPARISON  BETWEEN  INDUCTION  MOTORS  WITH  ROTORS 
SHORT-CIRCUITED  THROUGH  RINGS  OR  OF  THE  SQUIRREL- 
CAGE  TYPE;  AND  ROTORS  SHORT-CIRCUITED  THROUGH 
SYMMETRICAL  POLY-PHASE  BRUSHES 

The  e.m.f.  induced  through  rotation  in  a  rotor  of  the  slip-ring 
type  is  proportional  to  the  slip  and  it  is  of  slip-ring  frequency. 

The  e.m.f.  induced  through  rotation  in  a  rotor  of  the  commu- 
tator type  is  proportional  to  the  slip  but  it  is  of  primary  frequency. 

At  standstill  and  at  synchronism  it  appears  evident  that  both 
types  of  motor  must  operate  identically  if  we  consider  for 
example  two  three-phase  rotors,  or  a  rotor  with  slip-rings  on  one 
side  and  a  commutator  on  the  other,  alternately  short-circuited. 

Consider  for  a  moment  a  rotor  without  leakage  reactance  of 
any  kind.  The  mechanical  brush  shift  in  the  direction  of  rota- 
tion of  the  magnetic  field  leads  to  an  induced  e.m.f.  which  lags 
behind  the  e.m.f.  induced  if  the  brushes  are  in  a  position  in  which 
the  primary  and  secondary  entries  are  opposite  each  other. 
The  maximum  ampere  turns  on  the  rotor  occurring  thus  later  in 
time-phase,  the  rotor  position  being  advanced  in  space-phase, 
these  two  conditions  automatically  compensate  each  other  and 
thus  it  appears  that  it  is  immaterial  in  which  position  the  brushes 
are  placed,  the  same  argument  holding  if  the  brushes  are  shifted 
in  a  direction  opposite  to  that  of  the  rotation  of  the  magnetic 
field.1  The  method  of  proof  here  given  is,  of  course,  not  limited 
to  a  rotor  without  leakage,  as  the  phase  lag  of  the  current  re- 
mains unaltered  if  the  position  of  the  rotor  relative  to  the  stator 
does  not  affect  the  leakage  reactance  of  the  rotor. 

But  in  a  rotor  without  leakage  reactance,  perfect  reflection  of 
the  rotor  m.m.f.  into  the  stator  is  the  same  in  both  the  slip-ring 
type  and  the  commutator  type.  However,  with  secondary 

1  For  this  simple  method  of  exposition  I  am  indebted  to  my  friend, 
PROF.  V.  KARAPETOFF.  I  am  also  indebted  to  him  for  the  happy  term 
"perfect  reflection." 


POLY-PHASE  COMMUTATOR  MOTORS 


129 


leakage  reactance  it  appears  that  in  the  slip-ring  type  this 
reactance  is  proportional  to  the  slip  on  account  of  the  fact  that 
the  secondary  currents  are  of  slip  frequency,  whereas  in  the 
commutator  type  these  currents  are  of  primary  frequency.  It 
is,  therefore,  to  be  assumed  at  the  outset  that  the  two  types 
will  not  act  the  same  if  there  is  secondary  leakage  reactance. 

In  the  slip-ring  type,  both  the  effect  of  the  rotating  leakage 
field  $2(^2  —  1)  and  the  effect  of  single-phase  leakage  fields  are 
obviously  of  the  same  nature.  This  is  not  so  in  the  commutator 


Slip  Ring  Type    (Motor  Range) 


m 


Commutator  Type 
(Motor  Range) 


Slip  Ring  Type 
(Generator  Range 


Commutator  Type 
(Generator  Range) 


FIG.  97. — Comparison  of  commutator  type  of  induction  motor  with  the  slip  ring 
or  squirrel-cage  type. 

type  as  here  the  effect  of  a  rotating  leakage  field  $2(^2  —  1)  and 
the  effect  of  single-phase  leakage  are  altogether  different. 

The  assumption  of  constant  secondary  reactance  leads  to  a 
very  simple  diagram,  but  this  assumption  is  not  warranted. 
However,  to  bring  home  the  subject,  let  us  for  a  moment  con- 
sider the  logical  consequences  of  such  an  assumption.  In  Fig.  97 
F  is  the  common  resulting  flux  in  primary  and  secondary;  e2  = 
s.ei  is  the  e.m.f.  induced  by  rotation  in  a  group  of  coils  between 
brushes.  ^2  is  the  secondary  lag 

W.-^2  (124) 


130 


INDUCTION  MOTOR 


where  L2  is  the  secondary  leakage  inductance,  and  coiL2  is  the 
constant  secondary  leakage  reactance.  The  remainder  of  the 
diagram  is  familiar  to  the  reader.  Now,  as  ^2  is  constant  by  our 
assumption  of  constant  secondary  leakage  reactance,  the  angles 

at  "a"  and  "e"  are  equal  to  ~  +  ^2  and  therefore  e  moves  on  the 

£ 

TT 

arc  of  a  circle  so  that  angle  dem  is  always  equal  to  ^  +  ^2. 

& 

It  is  interesting  to  note  that  here  the  generator  range  is  greatly 
benefitted  by  the  use  of  the  commutator,  while  the  motor  range 
has  been  affected  detrimentally.  Both  types  have  the  point 
"A"  in  common  at  starting,  for  which  the  secondary  current 
and  phase  relations  are  identical. 

We  commend  to  the  reader  a  careful  study  of  the  diagram  and 
especially  the  curious  effects  above  synchronism,  to  which  atten- 
tion has  been  drawn  by  Riidenberg1  and  Altes,  and  which  are  so 
important  that  a  brief  reference  to  them  should  be  made  here. 

D.  THE    REFLECTION   INTO    THE    PRIMARY    CIRCUIT    OF    THE 
M.  M.  F.  OF  THE  SECONDARY  WITH  SLIP-RING  AND  COMMU- 
TATOR ROTORS 

At  positive  and  negative  slips,  the  angle  of  current  phase 
appears  reflected  differently  into  the  primary  if  the  slip  is  positive 
or  negative,  and  if  the  rotor  is  of  the  slip-ring  type  or  of  the 
commutator  type.  This  is  illustrated  in  the  following  table: 

SLIP-RING  TYPE 


+s 


1  R.    RUDENBERG,    E.  T.    Z.,  1910,    p.  1087. 
A.  I.  E.  E.  Trans.,  1918,  p.  309. 


Also    W.    C.    K.    ALTES, 


POLY-PHASE  COMMUTATOR  MOTORS 
COMMUTATOR  TYPE 


131 


—  1*2 


X. 

J- 


For  a  negative  slip,  which  corresponds  in  the  short-circuited 
types  to  generator  action,  we  see  therefore  that  in  the  slip-ring 
type  the  reflected  m.m.f.  of  the  secondary,  though  its  current 
lags  behind  the  induced  secondary  e.m.f.,  appears  in  the  primary 
vector  diagram  as  a  leading  current. 

In  the  commutator  type,  however,  the  reflected  m.m.f.  of  the 
secondary  appears  with  a  lag  due  to  the  fact  that  induced  e.m.f. 
and  induced  current  reverse,  while  the  reactance  component  is 
constant,  viz.,  tg\f/z  =  constant. 


E.  VARIABLE  AND  CONSTANT  SECONDARY  REACTANCE  OF  THE 
COMMUTATOR  MOTOR 

The  next  step  now  is  to  consider,  in  addition  to  the  constant 
secondary  reactance,  that  part  which  varies  with  the  slip.  We 
assume,  at  standstill,  point  A,  the  total  secondary  leakage  com- 
posed of  the  rotating  field  leakage  Bb  and  the  single-phase  leakage 
aB,  so  that  angle 

<BOa  =  tgti  =  const.  (125) 


Then  it  follows  from  Chap.  Ill,  B,  that  we  obtain  the  circle  0" 
for  the  locus  of  the  primary  current  of  a  commutator  type  of 
induction  motor  with  short-circuited  brushes,  while  the  circle  0' 
is  the  locus  for  a  squirrel-cage  or  slip-ring  type  of  motor. 

The  circle  is  larger  over  the  generator  range  due  to  the  fact 
that  the  part  of  the  secondary  leakage  reactance  which  is  constant 
appears  in  the  primary  as  would  a  leading  current  in  the  slip-ring 
type.  It  is  worth  while  to  follow  this  matter  up  with  a  physical 
illustration. 


132 


INDUCTION  MOTOR 


Let   Fig.    99   be  a  Leblanc  commutator.     A  rotating  field  F 
is  produced  by  the  currents  flowing  through  the  brushes,  its 


FIG.  98. — Variable    and    constant    secondary    reactance    of    the     commutator 

induction  motor. 

angular  velocity  being  coi.     If  o)2  >  coi  then  we  have  shown  that 
the  current  leads  the  impressed  e.m.f.     Now,  it  makes  no  differ- 


FIG.  99. — The  operation  of  the  Leblanc  commutator. 

ence  whether  F  is  produced  by  the  currents  from  the  line,  or 
whether  it  is  set  up  by  other  means,  if  the  brushes  are  short- 


POLY-PHASE  COMMUTATOR  MOTORS 


133 


circuited,  the  short-circuit  corresponding  to  the  negligible  impe- 
dance of  the  line,  as  was  so  brilliantly  pointed  out  by  M.  Latour. 
Thus  the  two  diagrams  of  Fig.  99  are  physically  the  same,  and  the 
effect  of  operating  a  commutator  motor  above  synchronism  consists 
in  creating  leading  currents  in  it,  as  viewed  from  the  stator,  and 
thus  the  constant  part  of  the  secondary  leakage  reactance  above 
synchronism  acts  like  a  condensance.  The  reader  is  cautioned 


Squirrel 

Cage 
or  Slip  Ring 

A.  UVW;/J  S)PS  ^     Induction 

Motor 


B. 


M 

c  < 
.   "53 

• 

c 

M 

CO 

o 

O    4, 

0-3 

O    O 
<^>    *"* 

"i     AAAAAAAA' 

Commutator 
Induction 
Motor 

lA^\Mf=^ 

YYVVVVVV 

c. 


wvwwvwv 


Commutator 
Induction 

Motor 

after  Arnold- 
Lacour) 


FIG.  100.  —  Commutator  and  slip  ring,  or  squirrel  cage,  types  of  induction  motor, 
and  their  equivalent  circuits. 

to  distinguish  between  the  total  reactance  and  the  leakage  react- 
ance, which  is  a  prolific  source  of  confusion. 

Equivalent  circuits  can  be  drawn  for  the  different  types  of 
induction  motors  as  is  indicated  in  Fig.  100,  where  "A"  is  the 
equivalent  circuit  of  the  slip-ring  type,  "B"  the  equivalent 
circuit  of  a  commutator  type  assuming  constant  secondary  react- 
ance, and  "C"  is  the  equivalent  circuit  for  a  commutator  type 


134 


INDUCTION  MOTOR 


in  which  both  constant  secondary  reactance  and  secondary  react- 
ance varying  with  the  slip  are  indicated. 

Tests  have  been  made  to  check  the  performance  of  these 
motors  by  E.  Arnold  and  la  Cour1  and  by  L.  Dreyfus  and  F. 


FIG.    101.  —  Slip-ring    type   of   induction   motor. 

chronism. 


Space   diagram   below    syn- 


Hillebrand.2 The  latter  equipped  a  rotor  with  slip-rings  on 
one  side  and  a  commutator  on  the  other  and  thus  recorded 
the  standard  circle  diagram  and  the  displaced  circle  for  the 
same  type  of  motor. 


-02 


FIG.    102. — Slip-ring    type    of    induction    motor.     Space    diagram   above   syn- 
chronism. 

In  Figs.  101 , 102, 103,  and  104  there  are  traced  out  the  m.m.f .  belts 
produced  by  the  rotation  of  the  resultant  field  F  with  relative  angular 

1  E.  ARNOLD  and  J.  L.  LA  COUR,  Vol.  V,  2,  p.  221. 

2  L.  DREYFUS  and  F.  HILLEBRAND,  "Zur  Theorie  des  Drehstromkollector- 
Nebenschlussmotors."    Elektrotechnik  und  Maschinenbau,  1910,  p.  886, 


POLY-PHASE  COMMUTATOR  MOTORS 


135 


velocity  coi  —  w2  towards  the  rotor.  For  o>i  >  o>2  the  machine  is  a 
motor,  while  f  or  coi  <  oj2  it  is  a  generator.  The  m.m.f .  belts  indicate 
that,  in  the  Slip-ring  Type,  viewed  from  the  primary,  the  m.m.f. 

of  the  secondary,  or  its  fictitious  flux  <£2,  lags  behind  F  by     +  ^2 


FIG.  103. — Commutator-type     of     induction     motor.     Space     diagram     below 

synchronism. 

electrical  degrees,  because  the  secondary  currents,  being  of  slip 
frequency  coi  —  co2,  set  up  a  rotating  magnetic  field  which  is 
carried  around  by  the  rotor  with  angular  velocity  W2  in  the 


FIG.  104. — Commutator-type    of    induction    generator.     Space    diagram    above 

synchronism. 

direction  of  mechanical  rotation.  Therefore,  the  electric  phase 
lag  of  the  secondary  current  appears  also  as  a  lag  in  the  com- 
bined space  and  time  diagram. 

For  the  generator  action  of  the  slip-ring  type  of  motor,  it  is 


136  INDUCTION  MOTOR 

to  be  noted  that  the  secondary  currents  set  up  a  magnetic  field 
rotating  in  opposition  to  the  mechanical  rotation  of  the  machine, 
thus  the  lagging  current  in  the  secondary  appears  in  the  primary 
and  in  the  combined  space  and  time  diagram  leading  the  flux 

F  by  a  time  or  space  angle  of  ~  +  ^2- 

z 

These  relations  are  different  in  the  Commutator  Type,  in  which 
the  secondary  currents  are  of  line  frequency  and  the  coil  groups 
between  brushes  are  stationary.  Therefore,  the  secondary 
current  reflected  into  the  primary  appears  as  a  lagging  current 
relative  to  the  induced  e.m.f.  Therefore,  this  consideration 
leads  again  to  the  curious  result  that,  in  the  Commutator  Type  of 
Induction  Generator  running  above  synchronism,  constant  sec- 
ondary reactance  strengthens  the  resultant  field  of  the  machine 
and  it  therefore  acts  as  capacity  does  in  the  range  below 
synchronism. 

F.  THE      SLIP-RING      COMMUTATOR     TYPE      AS     FREQUENCY 

CHANGER 

The  first  to  suggest  the  use  of  an  armature  provided  with  a 
commutator  on  one  side,  on  which  poly-phase  brushes  are  placed, 
and  slip-rings  on  the  other,  appears  to  be  Mr.  B.  G.  Lamme,1 
who  noticed  that,  in  a  rotary  converter  without  field  excitation, 
running  below  synchronism,  a  current  of  low  frequency  appeared 
at  the  brushes  on  the  commutator.  This  low  frequency  dis- 
appeared at  synchronous  speed. 

If  poly-phase  currents  of  frequency  ^i  are  sent  through  the 
brushes  upon  the  slip-rings,  then  a  magnetic  field  is  set  up 
rotating  with  angular  velocity  coi  relative  to  the  armature.  If 
the  armature  revolves  with  angular  velocity  co2  against  the 
direction  of  rotation  of  the  field,  then  in  the  groups  between  the 
stationary  poly-phase  brushes  upon  the  commutator,  there  will 
be  induced  an  e.m.f.  of  the  frequency  ^i  —  ^2.  Hence,  if  the 
commutator  of  such  a  device  were  connected  to  the  rotor  of  a 
slip-ring  type  of  induction  motor,  it  would  receive  currents  of 
slip  frequency  ^i  —  ^2,  and  on  its  slip-rings  it  would  deliver 
currents  of  the  frequency  ^i.  An  application  of  this  interesting 
phenomenon  is  described  in  Chap.  XIII,  C. 

1B.  G.  LAMME,  United  States  Patent  No.  682,943.  Sept.  17,  1901. 
Application  filed  July  24,  1897. 


CHAPTER  X 


THE  SERIES  POLY-PHASE  COMMUTATOR  MOTOR 

A.    THE    THEORY    FOR    CONSTANT    CURRENT   AND    CONSTANT 
POTENTIAL  IN  THE  IDEAL  MOTOR 

The  properties  of  a  commutator,  as  discussed  in  Chap.  IX, 
are  now  to  be  applied  to  the  Series  Poly-phase  Commutator 
Motor,  first  described  in  1888  by  Wilson  in  the  British  Patent 
No.  18,525  and  by  H.  Goerges  in  the  German  Patent  No.  61,951 
of  Jan.  21,  1891.  Mr.  H.  Goerges  also  described  the  Shunt 
Poly-phase  Commutator  Motor  and  outlined  the  theory  of  these 
motors  in  the  E.  T.  Z.,  1891,  p.  699. 

Figure  105  shows  diagrammatically  the  connections  of  the 
motor.  The  stator  windings, 
which  are  indicated  here  in 
F-connection,  are  in  series 
with  the  delta-connected  ar- 
mature which  is  rotating 
counter-clockwise  with  the 
angular  velocity  co2.  The 
rotating  field  resulting  from 
the  action  of  the  poly-phase 
currents  is  assumed  to  have 
a  counter-clockwise  rotation 
coi,  and  the  brushes  are  shown 
shifted  forward  by  an  angle  a 

in   the    direction    of    rotation.    Goerges).     Brush  shift  angle  a,  no  trans- 

The  neutral  position  of  the  former' 

brushes,  or  the  datum  from  which  we  count  the  brush-shift,  is 
defined  as  that  in  which  the  current  passing  in  series  through 
stator  and  rotor  produces  two  fictitious  magnetic  fields,  which 
cancel  each  other,  neglecting  leakage.  As  usual  we  assume  a 
two-pole  magnetic  structure  and  distributed  windings  in  rotor 
and  stator.  The  brushes  slide  on  the  commutator  here  assumed 
to  be  the  surface  of  the  rotor  wnidings. 

Great  attention  has  to  be  paid  to  the  conception  of  time  and 
space  phases.  To  simplify  the  analysis,  we  assume  identical 

137 


138 


INDUCTION  MOTOR 


windings  on  stator  and  rotor,  the  series  connection  being  obtained 
through  a  series  transformer  whose  magnetizing  current  and 
leakage  we  neglect  for  the  present,  Fig.  106. 

The  rotor  and  stator  currents  are  assumed  equal  and  in  time- 
phase,  so  that,  with  brush-shift  angle  a  =  0,  the  two  fictitious 
fields  of  rotor  and  stator  would  obliterate  each  other. 

If  the  stator  windings  acting  alone  produce  a  counter-clockwise 
or  positively  rotating  magnetic  field  <J>i,  then  the  same  current — 
equal  in  magnitude  and  time-phase — acting  alone  in  the  rotor 
windings,  produces  a  magnetic  field  $2,  whose  position  in  space 
at  a  given  moment  of  time  is  represented  by  the  vector  <J>2  in  Fig. 


FIG.  106.  FIG.  107. 

FIG.  106. — Three-phase  series  A.  C.  commutator  motor  with  series  transformer- 
Brush  shift  angle  a,  A-Connection  in  stator  and  rotor. 

FIG.  107. — Space  diagram  of  fluxes  in  series  motor  with  brush  shift  angle  a. 
Heavy  lines  indicate  space  vectors. 

107.  At  the  same  moment  of  time  the  same  current  produces  a 
magnetic  field  $1,  whose  position  in  space  is  indicated  by  the 
vector  3>i.  -The  vector  difference  of  3>i  and  $2  is  the  resultant 
really  existing  rotating  magnetic  field  F. 

With  a  =  0,  there  is  no  torque;  with  brush  shift  clockwise  or 
negative,  we  obtain  clockwise  rotation;  with  brush  shift  a 
counter-clockwise  or  positive,  we  obtain  counter-clockwise  rota- 
tion of  the  armature,  the  direction  of  rotation  reversing  with  the 
shift  of  the  brushes.  The  torque  is  exerted  in  the  direction  of  the 


SERIES  POLY-PHASE  COMMUTATOR  MOTOR 


139 


brush  shift  from  the  defined  datum  a  =  0,  as  is  indicated  by  a 
simple  consideration  of  the  magnetic  fluxes  <t>i  and  $2  and  their 
mutual  attraction.1 

Still  assuming  no  leakage,  we  know  that  F  induces  in  stator  and 
rotor  windings  e.m.fs.  which  are  in  time  quadrature  with  the 
resultant  flux  which  embraces  the  windings.  As  the  e.m.f. 
induced  in  the  rotor  windings  must  ap- 
pear earlier  in  time-phase  than  that 
induced  in  the  stator  windings,  with  a 
negative,  or  later  with  a  positive,  it 
follows  that  the  e.m.fs.  induced  in  the 
stator  and  rotor  differ  in  time-phase  by 
the  same  time  angle  a  as  do  the  space 
fields  3>i  and  <£2  by  the  same  space  angle  a. 

Assume  the  resultant  flux  F  to  be  pro- 
jected on  a  vertical  time  axis  for  refer- 
ence (Fig.  108).  Assume  it  to  be  zero  at 
a  certain  time.  Then  the  voltage  Ea  in- 
duced in  the  stator  winding  of  the  series 
motor  is,  barring  leakage  and  resistance, 
in  quadrature  with  the  resultant  field  F  \ 
and,  as  the  current  /  is  in  time-phase  with  \ 

the  fictitious  flux  $1,  we  now  have  the        .    \ 
essential  elements  for  the  determination 
of   the  complete  vector  diagram  of  the       FlG      i08.- Combined 

poly-phase  Series  motor.  space  and  time  diagram  of 

Assume   the   rotor   and  stator  resist-    X^^co—tr 
ances  zero,  and  the  rotor  standing  still,    motor    with    brush   shift 
The  fictitious  fluxes  $1  and  <J>2  differing    angle  a' 
in  space  phase  by  the    angle    a,    produce   the    resultant   real 
field  F. 

I  is  in  time-phase  with  3>i,  and  Ea  is  in  time  quadrature  with  F. 
Thus  the  time  lag  \f/a  of  the  current  /  behind  Ea  is  determined. 
Ea  and  Eb  are  in  time-phase  opposition  if  the  brush-shift  angle 
a  =  0.  They  differ  by  the  angle  a  in  time-phase.  We  thus 
obtain  E  as  the  resultant  voltage  impressed  upon  the  motor,  in 
time  quadrature  with  J,  in  the  case  of  a  resistance-less  imaginary 
motor.  This  relation  follows  from  the  similar  m.m.f.,  or  flux, 
and  e.m.f.  triangles. 

Assuming  /  constant  and  the  motor  beginning  to  turn  in  the 

1  See  V.  KARAPETOFF,  "The  Secomor,"  Trans.  A.  I.  E.  E.,  Feb.  16,  1918. 


140 


INDUCTION  MOTOR 


direction  of  its  rotating  field,  then  the  counter  e.m.f.  of  rotation 
induced  in  the  rotor  diminishes  proportionally  to  the  slip.  It 
disappears  at  synchronism. 

An  examination  of  the  diagram  Fig.  108  shows  that,  below 
synchronism  the  stator  takes  energy  from  the  line,  while  the  rotor 
delivers  energy  back  to  it.  At  synchronism  the  stator  alone 
takes  energy  from  the  line ;  above  synchronism,  it  is  seen  that  both 
stator  and  rotor  take  energy  from  the  line,  the  sum  being  trans- 
formed into  mechanical  energy.  Such  a  motor  is  therefore 

described  as  "doubly-fed,"  a  term 
widely  used  in  the  great  work  of  the 
joint  authorship  of  E.  Arnold,  J.  L. 
la  Cour,  and  A.  Fraenckel. 

We  have  already  seen  that  the  time- 
phase  of  Eb  depends  upon  the  brush 
shift  a  which  determines  the  space 
angle  of  the  m.m.fs.  To  emphasize  this 
important  point  once  more,  for  a  =  0, 
Ea  and  Eb  are  in  time-phase  oppo- 
sition, Fig.  109.  Shift  the  brushes 
clockwise,  or  backward  in  the  direc- 
tion of  the  rotation  of  the  magnetic 
field,  and  the  maximum  of  Eb  will 
occur  earlier  in  time-phase  by  the 
angle  a,  which  is  now  a  time  angle; 


FIG.  109.— Brush  shift  and 
field  rotation  affect  the  time 
diagram. 

shift  the  brushes  counter-clockwise, 

or  forward  in  the  direction  of  rotation  of  the  magnetic  field,  and 
the  maximum  of  Eb  will  occur  later  in  time-phase  by  the  angle  a. 
We  shall  consider  only  this  latter  case  in  which  the  rotor  and  the 
magnetic  field  rotate  in  the  same  direction. 

Still  neglecting  leakage  and  resistance,  we  re-draw  (Fig.  110)  the 
time  diagram  of  the  e.m.f  s.  and  of  the  current  and  we  see  at  a  glance 
that  for  constant  current  the  e.m.f.  triangle  is  OBC,  in  which  OB 
is  the  e.m.f.  induced  in  the  primary  or  stator  winding,  BC  is 
the  e.m.f.  induced  in  the  rotor  at  slip  s,  and  it  is  therefore  under 
our  previous  assumption  of  equal  numbers  of  turns  equal  to  Ea. 
The  point  B,  therefore,  corresponds  to  synchronous  rotation  and 
the  range  BA  corresponds  to  speed  above  synchronism. 

The  angle  a  at  B  remains  constant  for  a  full  speed  range,  and 
so  does  the  angle  a  —  \l/a  at  A.  If,  therefore,  we  were  to  keep 
OC  =  E  constant,  as  well  as  a,  allowing  the  current  to  vary,  it  is 


SERIES  POLY-PHASE  COMMUTATOR  MOTOR 


141 


obvious  that  A  lies  on  the  periphery  of  a  circle  described  about 
OC  as  chord.  The  current  7  is  always  proportional  to  OA, 
therefore,  I  may  be  measured  by  the  chord  drawn  from  0  to  the 
periphery  of  the  circle.  (Fig.  111.) 

The  Torque. — The  current  at  starting  is  ASh,  while  the  torque 
at  starting  is  proportional  to  the  product  of  the  space  quadrature 
component  of  the  secondary  m.m.f.  into  the  magnetic  flux  F,  or 


FIG.  110.— "D,"    Starting;    "B,"    Synchronism;    "A,"    Double    Synchronism. 
Time-phase  diagram  for  the  locus  of  the  primary  current  in  the  series  poly-phase 
commutator  induction  motor  for  constant  potential.     Currents  measured  from 
0  to  periphery  of  circle. 


Torque  =  K-IF-  sin 


(126) 


But  in  our  diagram  F  is  proportional  to  and  in  time  quadrature 
with  Ea,  and  Ea  is  also  proportional  to  7.     Therefore, 


a 


Torque  =  Kil2  sin  ^ 


(127) 


The  torque,  therefore,  may  be  represented  by  the  square  of  the 
current  and,  as  we  have  seen  on  previous  occasions,  the  square  of 
the  chords  in  a  circle  may  be  measured  graphically  perpendicular 
to  the  diameter  of  the  circle  as  shown  in  Fig.  112  by  T. 


142 


INDUCTION  MOTOR 


The  Slip. — Drop  a  perpendicular  from  Aay,  the  current  locus 
at  synchronism,  upon  the  diameter  of  the  circle  OAQ.  (Fig. 
113.)  The  point  of  intersection  a  between  OA  and  AsyS,  cuts 


FIG.  111. — Circle   OAshA   is   locus   of   current   I   for   constant   E  =  OC.     The 
current  and  the  e.m.f.'s  in  the  ideal  series  motor  for  constant  potential. 


FIG.  112. — The  torque  in  series  poly- 
phase commutator  motor.  OC  =  E  = 
primary  impressed  voltage.  For  ^i  = 
0,  OC  =  I  and  T  is  corresponding  torque. 


FIG.  113.— The  slip  in  the  series 
poly-phase  commutator  motor  for 
constant  potential. 


off  ASJ/a,  which  is  a  measure  of  the  slip.     Point  S  corresponds  to 
100  per  cent  slip,  point  A8y  to  a  slip  0  per  cent. 


SERIES  POLY-PHASE  COMMUTATOR  MOTOR 


143 


Proof. — Triangles  ObC  and  OAsya  are  similar.     The  slip  is  the 
ratio  of  Cb  :  Ob,  therefore, 

Cb  :0b  ::aAsy  :  OAsy 
.   aA8V 


OA 


sy 


(128) 


At  double  synchronism,  as  is  shown  by  inspection  of  the  dia- 
gram, the  power  factor  becomes  unity. 


B.  THE  THEORY  FOR  CONSTANT  CURRENT  AND  CONSTANT 
POTENTIAL  IN  THE  REAL  MOTOR 

The  diagram  of  m.m.fs.,  or  fluxes,  remains  the  same  if  we  take 
the  leakage  into  consideration  by  using  the  e.m.f.  induced  by  the 


FIG.  114. — The  time-phase  diagram  of  the  series  poly-phase  commutator  induc- 
tion motor.     Including  resistance  and  leakage. 

leakage  field  which  is  the  simplest  method  in  this  case  as  we  must 
combine  e.m.fs.  affected  by  the  speed.  In  the  theory  of  the  slip- 
ring  or  squirrel-cage  induction  motor,  we  found  it  simpler  to 
employ  leakage  fields  instead  of  the  voltages  induced  by  them. 


144 


INDUCTION  MOTOR 


The  leakage  reactance  of  the  stator  windings  is  constant  at 
all  speeds.  The  leakage  reactance  of  the  rotor  windings,  as  shown 
in  Chap.  IX,  is  composed  of  a  part  constant  at  all  speeds  and  of 
another  dependent  upon  the  speed  of  rotation  of  the  armature. 
This  latter  is  positive  at  speeds  below  synchronism,  zero  at  syn- 
chronism, and  negative  at  speeds  above  synchronism.  Without 
a  serious  error  we  may  permit  ourselves  the  license  of  viewing 
the  total  leakage  reactance  of  the  motor  as  constant,  and  in 


Power 


Torque 


100% 
80% 


1  .8  .6  .4  .2  0 

Slip 

FIG.  115. — The  torque,  current,  power  factor  and  slip  in  the  series  poly-phase 
commutator  induction  motor. 

quadrature  time-phase  with  the  current,  Fig.  114.  Adding  vec- 
torially  to  the  e.m.f.  thus  obtained  /(ri  +  r2),  a  point  G  is  derived 
at  the  intersection  of  OA  and  the  line  OH  through  H.  The 
angle  (a  —  \f/a)  at  G  is  again  constant. 

If  we  assume  again  constant  primary  voltage  OH  for  the  entire 
operating  range,  a  circle  on  whose  periphery  lie  the  points  0, 
G,  and  H  becomes  the  locus  of  a  radius  vector  from  0,  like  OG, 
which  is  a  measure  of  the  current  /. 


SERIES  POLY-PHASE  COMMUTATOR  MOTOR  145 

All  other  relations  follow  from  this  diagram  as  before.  We 
have  plotted  in  Fig.  115  in  rectangular  coordinates  the  current, 
the  power  factor,  the  torque,  and  the  power  as  a  function  of 
the  rotor  slip.  At  speeds  above  synchronism  the  power  factor 
approaches  unity.  It  may  be  suggested  to  the  reader  to  draw 
similar  diagrams  for  different  brush-shifts  a. 

C.  THE  NECESSITY  OF  SATURATION  FOR  STABILITY 

The  series  poly-phase  motor  is  self-exciting  as  a  generator. 
If  it  possesses  remnant  magnetism,  it  may  generate  direct  cur- 
rent as  the  supply  circuit  forms  virtually  a  short  circuit  for  such 
currents.  Saturation  of  the  motor  frame  or  of  the  series  trans- 
former may  prevent  these  effects  to  a  great  extent.  On  this 
subject  the  reader  may  consult  the  following: 

U.  S.  Patent  1,164,223,  Dec.  14,  1915,  A.  SCHERBUJS:  Stabilized  Commu- 
tator Machine. 

E.  ARNOLD,  J.  L.  LA  COUR  and  A.  FRAENCKEL,  Die  Wechselstromkom- 
mutatormaschinen,  1912,  p.  59. 

V.  KARAPETOFF,  "The  Secomor."  Trans.  A.I.E.E.,  1918,  Vol.  XXXVIII, 
Part  1,  p.  347. 

W.  C.  K.  ALTES,  "The  Poly-phase  Shunt  Motor."  Trans.  A.  I.  E.  E., 
1918,  Vol.  XXXVII,  Part  1,  p.  385. 


10 


CHAPTER  XI 
THE  SHUNT  POLY-PHASE  A.  C.  COMMUTATOR  MOTOR 

A.  HISTORICAL  INTRODUCTION 

The  shunt  poly-phase  commutator  motor  as  well  as  the  series 
type  appear  to  be  the  invention  of  H.  Goerges  who  described  them 
in  the  E.  T.  Z.,  1891,  p.  699.  The  10  years  which  succeeded  its 
invention  were  devoted  to  the  practical  development  of  the  Tesla 
induction  motor  and  thus  the  commutator  type  did  not  receive 
much  attention.  Exactly  10  years  after  Georges'  publication,  A. 
Heyland  described  again  (in  the  E.  T.  Z.,  1901,  No.  32),  the 
Goerges  motor,  showing  that  it  can  be  used  to  compensate  the 
watt-less  component  of  the  primary  by  proper  brush-shift. 
Although  Prof.  Blondel  contends1  that  this  was  a  matter  of 
course,  it  would  have  been  an  interesting  contribution  had  not 
Mr.  Heyland  claimed  with  great  emphasis  that  his  motor  was 
entirely  different  in  principle  from  the  shunt  motor  of  Goerges. 
This  has  been  disproved  with  great  precision  and  clarity  by  Prof. 
Blondel  in  the  papers  cited.  Mr.  Heyland  had  suggested  the  use 
of  stationary  sliding  contacts  on  the  squirrel-cage  rings,  thus 
introducing  an  external  e.m.f.  at  points  equally  spaced  on  the 
commutator,  but  the  low  resistance  of  these  end  rings  acts  as 
a  powerful  shunt  and  this  arrangement  proved  ineffective.  The 
author  tried  two  independent  windings  with  somewhat  better 
success,  and  tests  on  a  similar  motor  are  reported  by  Prof.  C.  A. 
Adams.2  It  is  now  no  longer  open  to  doubt  that  Mr.  Heyland's 
suggestion  covers  solely  a  shunt  poly-phase  motor  with  addi- 
tional shunts  placed  between  the  commutator  bars.  No  practical 
application  seems  to  have  been  made  of  this  modification. 

Great  activity  in  devising  modifications  and  improvements  of 
poly-phase  commutator  motors  followed  the  general  enthusiasm 
created  by  Mr.  B.  G.  Lamme's  single-phase  railway  motors.  The 
use  of  these  motors  in  order  to  obtain  speed  regulation  without  un- 
due loss  in  efficiency,  and  finally  their  application  to  the  speed  regu- 

1  ANDRE  BLONDEL,  Theorie  des  Alternomoteurs  Poly-phas6s  &  Collecteur. 
L'Eclairage  Electrique,  1903,  pp.  121  to  495. 

2  C.  A.  ADAMS,  Trans.  A.  I.  E.  E.,  1903. 

146 


SHUNT  POLY-PHASE  A,  C.  COMMUTATOR  MOTORS     147 


lation  of  large  induction  motors  with  which  they  are  concatenated, 
has  secured  for  the  Goerges  motor  a  wide  and  interesting  field. 

In  order  to  obtain  the  appropriate  voltage  on  the  rotor  of  a 
Goerges  shunt  poly-phase  A.  C.  commutator  motor,  it  is  necessary 
either  to  use  a  separate  transformer,  or  to  utilize  the  stator 
winding  by  means  of  taps,  or  to  employ  a  regulating  winding. 
These  methods  are  treated  in  Chap.  XIII  where  the  work  of 
Osnos,  La  Cour,  and  Schrage  is  given  consideration. 

B.   THE  THEORY  OF  THE  SHUNT  POLY-PHASE  A.   C.  COMMU- 
TATOR MOTOR  FOR  CONSTANT  POTENTIAL 

Assume  a  stator  like  that  of  a  standard  induction  motor  in 
which  a  rotor  is  mounted  wound  like  a  direct-current  armature 
equipped  with  a  commutator.  Let  poly-phase  current  be  supplied 
to  both  the  rotor  and  the  stator  from  the  same  supply  circuit. 

We  thus  obtain  a  "  doubly- 
fed''  type  of  poly-phase 
motor,  which  is  called  a  shunt 
poly-phase  A.  C.  commutator 
motor. 

The  two  e.m.fs.  in  the  stator 
and  rotor,  being  derived  from 
the  same  supply  circuit,  are 
of  the  same  frequency  and  in 
time-phase. 

If  the  rotor-brush  position 
is  such  that  the  entries  on 
both  primary  and  secondary 
are  opposite  each  other,  then 
the  m.m.f.  belt  of  the  sec- 
ondary in  relation  to  that 
of  the  primary  depends  solely 
upon  the  time-phase  of  the 
two  circuits. 

If  the  rotor  brushes  are 
shifted  the  brush-shift  angle  displaces  the  impressed  e.m.f.  of 
the  rotor  relative  to  the  stator  by  the  amount  of  the  angle  of 
shift.  Thus,  in  Fig.  116,  let  E\  be  the  impressed  e.m.f.  on  the 
stator,  then,  if  a  is  the  brush  shift,  #2,  the  impressed  e.m.f.  on  the 
rotor  appears  displaced  by  the  angle  a  relative  to  the  impressed 
e.m.f.  EI  so  far  as  space  relations  are  concerned.  That  is  to 


FIG.  116. — Composition  of  e.m.f's.  in 
the  rotor  of  the  shunt  poly-phase  com- 
mutator motor. 

Ei  =  e.m.f.  impressed  on  stator. 

Ei  =  e.m.f.  impressed  on  rotor. 

62    =  e.m.f.  induced  in  rotor.  . 


148 


INDUCTION  MOTOR 


say,  the  m.m.f.  belts  due  to  the  currents  which  are  produced  by 
EI  and  Ez,  must  be  determined  as  though,  in  a  stationary  trans- 
former, the  two  e.m.fs.  were  displaced  in  time-phase  by  the 
angle  a. 

If  we  assume  for  the  moment  no  secondary  leakage,  then  the 
resultant  flux  F  induces  an  e.m.f .  e2  through  the  relative  angular 
velocity  o>i  —  co2  of  the  rotor  in  respect  to  F,  and  the  vector  sum 
of  E2  and  e2  results  in  Es,  which  produces  a  current  which  would 
be  in  phase  with  Es  if  there  were  no  leakage  in  the  secondary. 
If  there  is  leakage,  /2  would  lag  behind  E3  by  a  time  angle  ^2- 


m 


FIG.  117. — Time-phase  vector  diagram  of  the  shunt  poly-phase  commutator 
induction  motor.  Secondary  current  Is  is  composed  of  currents  .BCdue  to  e2  and 
AC  due  to  Ei.  I\  =  OB,  I3  =  BA,  i0  =  OA. 

The  phase  relation  of  EI  and  E%  depends  solely  upon  the  brush 
position,  if  they  are  derived  from  the  same  supply  circuit.  It 
is  difficult  but  important  to  bear  in  mind  that  we  are  tracing 
rotating  fields  in  space  and  that  time  and  space-phases  are  utilized 
in  the  same  diagram  so  that  we  can  show  the  mutual  effect  of 
rotor  and  stator  m.m.fs.  in  one  diagram  instead  of  in  two.  This 
may  be  confusing,  but  the  other  treatment  loses  in  physical 
reality. 


SHUNT  POLY-PHASE  A.    C.    COMMUTATOR   MOTOR     149 

The  m.m.f.  of  the  rotor  dueto/3andthem.m.f.  of  thestatordue 
to  1 1  result  in  the  magnetizing  m.m.f.  equivalent  to  i0  in  the  stator 
windings  (Fig.  117).  Assuming  constant  secondary  reactance  and 
neglecting  as  a  good  approximation  that  part  of  the  secondary  reac- 
tance which  is  proportional  to  the  slip,  /3  is  composed  of  BC  due  to 
e2,  and  of  AC  due  to  E^.  Point  C,  therefore,  is  a  fixed  point  as 
long  as  the  brush-shift  angle  a  and  E2  remain  constant.  Point 
m  also  is  a  fixed  point,  Od  :  dm  being  equal  to  vi  —  1,  as  is  readily 

seen.     Angle  CBm  is  equal  to  ~  +  ^2  and  constant  so  that  point 

B  moves  on  the  arc  of  a  circle  described  over  mC.  The  circle 
described  about  0'  as  center  is  the  primary  locus  of  the  commu- 
tator-induction motor  with  short-circuited  brushes.  The  circle 
described  about  0"  as  center  is  the  locus  of  the  stator  current  of 
the  shunt  poly-phase  motor,  to  which  has  to  be  added,  or  from 
which  has  to  be  subtracted,  vectorially,  the  current  in  the  pri- 
mary of  the  transformer  feeding  the  rotor.  (Fig.  118.) 


C.  DETERMINATION  OF  THE  TOTAL  PRIMARY  CURRENT 

With  the  limiting  assumption  of  a  constant  secondary  lag  \f/z 
between  the  secondary  total  e.m.f.  E%  and  the  total  secondary 
current  Is  we  may  now  proceed  to  determine  the  total  current 
taken  from  the  line  supplying  both  stator  and  rotor.  (Fig.  118.) 

The  stator  current  is  OB. 

The  rotor  current  is  Bdv\.  This  current  being  fed  into  the 
rotor  at  the  voltage  E2  which  is  smaller  than  EI,  if  the  ratio  of 
transformation  is  n,  then  the  current  to  be  added  to  the  stator 
current  on  account  of  the  current  /3  fed  into  the  rotor  from  #2 

is  — ,  to  be  added  to  /i  in  such  a  manner  that,  as  outside  the  motor 

Yl 

Ez  and  EI  are  in  time-phase,  the  phase  lag  between  73  and  EI 
must  be  the  same  as  that  between  73  and  E2.  Thus  results  a 

simple  graphical  method1  which  sets  off  Bh  =  —  at  the  constant 

71 

brush-shift  angle  a,  triangle  dBh  for  all  secondary  currents  being 
similar,  angle  dBb  always  being  a  and  angle  Bdh  being  c,  also 
constant.  Thus  draw  dO",  make  triangle  dO"Of"  similar  to 

1  A.  BLONDEL,  UEdairage  Electrique,  1903,  p.  178. 


150 


INDUCTION  MOTOR 


triangle  dBh,  and  0'"  is  the  center  of  the  new  circle  for  the  total 
current. 

This  total  current,  as  was  to  be  expected  in  view  of  the  double 
feeding  of  this  motor  through  its  primary  and  secondary,  is 
smaller  than  the  stator  current  over  the  motor  range  of  the  shunt 
machine  and  it  may  be  seen  at  a  glance  also  that  the  rotor 
returns  energy  to  the  supply  circuit.  The  whole  arrangement 


Stator 


E 


FIG.  118. — Time-phase  vector  diagram  of  the  shunt  poly-phase  commutator  in- 
duction motor.     Circle  loci  of  the  stator  current  and  of  the  total  current. 

may  be  simulated  by  a  sort  of  equivalent  arrangement  of  e.m.fs., 

resistances,  and  reactances,  as  is  shown  in  Fig.  119,  where  EI 
pi 

and  -  -  are  mechanically  coupled  together,  spaced  apart  by  a 
s 

space  angle  a.     In  order  to  obtain  similarity  of  current  and  e.mJ: . 
relations,  it  is  necessary  to  divide  Ez  by  the  dip  s. 


SHUNT  POLY-PHASE  A.   C.   COMMUTATOR  MOTOR     151 


D.  SPEED  REGULATION  AND  THE  SLIP 

It  is  evident  that,  since  BC,  Fig.  1 17,  represents  the  secondary 
current  due  to  e%  only,  and  as  62  is  proportional  to  the  product 
of  the  slip  into  F,  tgCmB  is  a  measure  of  the  slip.  By  changing 
E2  and  a  any  slip  may  be  obtained  and  thus  the  poly-phase  shunt 
motor  acts  in  this  respect  entirely  differently  from  the  induction 
motor,  the  shunt  motor  having  three  degrees  of  freedom,  while 
the  induction  motor  has  only  one.1  A  counter  e.m.f.  may  be 
injected  into  the  rotor  of  such  magnitude  and  phase  that  the 


L± 

s 


FlG.  119. — Equivalent  electrical   and   mechanical   combination   simulating   the 
action  of  the  shunt  poly-phase  commutator  induction  motor. 


motor  will  run,  say,  at  20  per  cent  slip,  of  which  say  4  per  cent  is 
ohmic  drop,  while  the  remaining  16  per  cent  is  due  to  the  in- 
jected e.m.f.  which  may  be  made  in  phase  with  the  ohmic  drop. 
However,  incidentally,  the  phase  of  the  injected  e.m.f.  may  be 
shifted  so  as  to  magnetize  the  motor  and  thus  to  supply  through 
the  secondary  the  magnetizing  current  ordinarily  supplied 
through  the  primary.  As  this  can  be  done  with  much  less 
K.  V.  A.  due  to  the  low  voltage  of  the  rotor  caused  by  the  high 
slip,  it  appears  obvious  that  such  a  motor  may  have  a  very  high 
power  factor. 

1  This  likeness  to  problems  in  dynamics  is  due  to  PROF.  V.  KARAPETOFF. 


152  INDUCTION  MOTOR 

E.  BIBLIOGRAPHY 

Before  leaving  the  subject  of  these  interesting  motors,  a  short  list  of  papers 
may  be  given  chronologically. 

The  motor  was  described  in  E.  T.  Z.,  1891,  p.  699,  by  H.  GOERGES,  by 
whom  it  was  also  patented  Jan.  21,  1891,  in  the  German  patent  No.  61,951. 

It  was  brought  back  to  light  10  years  later  largely  through  the  sensational 
paper  by  A.  Heyland,  E.  T.  Z.,  1901,  No.  32,  in  which  the  author  described 
the  same  doubly-fed  motor,  calling  it,  however,  an  induction  motor  excited 
from  the  secondary.  Through  shifting  the  brushes  any  primary-phase 
angle  may  be  obtained.  No  tests  were  made  or  described. 

A  fundamental  advance  was  made  in  the  theory  of  these  motors  by 
PROF.  A.  BLONDEL,  in  L'Eclairage  Electrique,  Apr.  25,  1903  et  seq.,  where, 
however,  a  curious  assumption  was  made.  PROF.  BLONDEL  assumes  that 
to  the  secondary  impressed  voltage  £'2  the  rotor  offers  resistance  only. 

ET 

He  thus  composes  —  and  iz  into  /a  and  then  applies  to  this  current  the 

flux  theory  of  the  induction  motor  assuming  a  secondary  leakage  field  to  be 
produced  by  this  m.m.f.  His  results  are  thus  marred  by  this  assumption, 
which  also  implies  that  the  secondary  leakage  lag  diminishes  with  the  slip 
of  the  motor,  which  is  at  best  only  partially  true  and  which  leads  to  the 
establishment  of  a  specious  equivalence  between  the  squirrel-cage  or  slip- 
ring  motor  and  the  commutator-induction  motor  short-circuited  across  its 
brushes,  a  result  which  the  tests  do  not  seem  to  bear  out.  BLONDEL  is  thus 
led  to  semi-circular  loci  where  we  have  arrived  at  arcs.  The  value  of 
BLONDEL'S  methods  is  fortunately  in  no  way  impaired  by  these  assumptions. 

Three  months  after  the  appearance  of  the  BLONDEL  circle  diagrams  of  this 
motor  in  Apr.  25,  1903,  MR.  HEYLAND  published  in  the  E.  T.  Z.  No.  30, 
July  23,  1903,  a  diagram  identical  with  BLONDEL'S  diagram  in  spite  of  the 
curious  assumption  made  by  BLONDEL.  No  reference  whatever  appears 
to  have  been  made  by  MR.  HEYLAND  to  BLONDEL'S  papers  here  referred  to. 

An  exhaustive  study  of  the  shunt  motor  and  the  derivation  of  a  cor- 
rect diagram  appeared  in  the  E.  T.  Z.,  1903,  p.  368  et  seq.,  by  PROF.  O.  S. 
BRAGSTAD. 

M.  EDOUARD  ROTH,  of  Belfort,  France,  published  a  masterly  thesis  in 
L'Eclairage  Electrique,  April  to  June,  1909. 

The  work  of  DR.  E.  KITTLER  and  DR.  W.  PETERSEN,  Stuttgart,  F.  ENKE, 
devotes  a  great  deal  of  space  to  these  motors. 

Vol.  V,  Part  2,  of  E.  ARNOLD,  LA  COUR  and  FRAENCKEL,  Berlin,  J. 
SPRINGER,  1912,  is  a  mine  of  valuable  information  on  alternating-current 
commutator  machines  in  general. 

DR.  F.  EICHBERG'S  "Gesammelte  Elektrotechnische  Arbeiten,"  1897- 
1912,  Berlin;  J.  SPRINGER,  1914,  may  also  be  consulted. 

The  papers  by  L.  DREYFUS  and  F.  HILLEBRAND  in  Elektrotechnik  & 
Maschinenbau,  1910,  pp.  367  et  seq.,  may  be  consulted  with  profit. 

The  latest  contributions  are  the  papers  by  N.  SHUTTLE  WORTH,  "Poly- 
phase Commutator  Machines  and  their  Application,"  The  Journal  of  the 
Institution  of  Electrical  Engineers,  Mar.,  1915,  arid  the  paper  by  W.  C.  K. 
ALTES,  "The  Polyphase  Shunt  Motor,"  Trans.  A.  I.  E.  E.,  1918. 


CHAPTER  XII 
METHODS  OF  SPEED  CONTROL 

A.  CONCATENATION 

The  similarity  in  theory  between  an  induction  motor  and  a 
transformer  is  due  to  the  fact  that,  as  the  secondary  frequency 
in  the  rotor  of  the  induction  motor  varies  from  full  primary  fre- 
quency at  standstill  to  zero  at  synchronism,  the  constant  resist- 
ance of  the  rotor  is  in  effect  equivalent  to  a  variable  resistance 
r-2,  -5-  s  in  the  secondary  of  a  transformer,  where  r%  is  the  second- 
ary resistance  of  the  rotor  and  s  the  slip,  viz.,  the  difference 
between  primary  and  secondary  frequencies  divided  by  the 
primary  frequency.  Thus  we  obtain  the  conception  of  the 
equivalent  circuits  which  simulate  the  physical  phenomena  of 
magnetizing  current,  leakage  fields,  etc. 

If  the  induced  e.m.f.  in  the  secondary  of  one  motor  at  the 
frequency  of  its  slip  is  impressed  upon  a  second  motor  (we  will 
assume  it  to  be  impressed  upon  the  stator  with  a  winding  having 
a  number  of  conductors  equal  to  that  of  the  rotor  of  the  first 
motor),  then  the  second  motor  will  not  operate  like  a  standard 
induction  motor  of  constant  impressed  voltage  and  constant 
frequency,  but  both  its  impressed  voltage  and  frequency  will 
vary  in  a  peculiar  manner.  If  the  rotors  of  both  motors  are 
mounted  rigidly  on  the  same  shaft,  then  they  have  a  common 
mechanical  angular  velocity  o)2.  We  thus  obtain  the  following 
relations  assuming  the  same  number  of  poles  in  both  motors  ;  and 
designating  the  angular  velocity  of  the  primary  field  of  Motor  I 
by  coi,  its  slip  by  81,  and  the  slip  of  Motor  II  by  sn. 

Sl=^^-°  (129) 

0)1 

But  o)2  =  o)i(l  -  si)  (130) 

The  relative  angular  velocity  at  which  the  rotor  conductors  of 
Motor  II  cut  through  the  field  impressed  by  the  e.m.f.  of  the 
secondary  of  Motor  I  is  on  —  o)2  —  w2  =  «i  —  2co2. 
Therefore,  the  slip  of  Motor  II, 


—  20)2 


0)1    — 

153 


/1Q1\ 

(131) 


154  INDUCTION  MOTOR 

And,  because  of  equation  (130), 

coi  —    2coi(l  —  Si) 
-    —  -  - 


/iQ<v\ 

v  (132) 

—    Sj) 


Sl  =      zn  (134) 

si«ii  =  2si  -  1  (135) 

These  relations,  which  are  here  obtained  only  for  equal  numbers 
of  poles  in  both  motors,  are  of  fundamental  importance  in  under- 
standing the  operation  of  concatenation  or  cascade  operation  of 
induction  motors. 

Beginning  the  examination  from  the  secondary  of  Motor  II, 
the  relative  angular  velocity  between  its  rotor  conductors  and 
its  resultant  rotating  field  is  <oi  —  2oo2.  Therefore,  the  e.m.f., 
induced  in  this  winding,  can-  be  written  as  follows  : 

64  =  2.12(~i  -  2^2)2^40-*  volts  (136) 

64  =  2.12  (~l  -  2~2)  (~*  ~  ~2)  -x^lO-*  volts     (137) 

\f^1    —       /^/2/    \          /^  / 

e4  =  2.12(sIIsI)~i24F410-8  volts  (138) 

In  the  equivalent  circuits  with  constant  frequency  ^i  the  resist- 
ance which  is  to  be  the  equivalent  of  r4,  the  rotor  resistance  of 
Motor  II,  in  which  the  frequency  is  equal  to  (^i  —  2~z),  is 
therefore  obtained  by  dividing  r4  by  fesn). 

The  effect  of  the  leakage  fields  remains  of  course  unaffected 
by  the  different  frequencies  in  the  different  parts  of  the  conca- 
tenated circuits. 

The  resistances  of  the  primary  of  Motor  II  and  the  secondary 
of  Motor  I  are  situated  in  circuits  of  variable  frequency  (~i  — 
~2)t  therefore,  the  e.m.f  s.  to  overcome  these  are: 


e3  =  2.12(~i  -  ~2)23^310-8  volts  (139) 

e2  =  2.12(~i  -  ~2)Z2F2W-B  volts  (140) 


These  equations  may  be  written: 

e3  =  2.12  /~i  ~  ~ 

\      '^•'i 

e2  =  2.12  (~*  ~  ~    ~iZ2F210-8  volts 


METHODS  OF  SPEED  CONTROL  155 

or  e3  =  2.12(sI)~1z3F310-8  volts  (141) 

e2  =  2.12(sI)~iZ2F210-8  volts  (142) 

In  other  words,  in  the  equivalent  electric  circuits,  simulating 
the  circuits  of  concatenation,  the  e.m.fs.  impressed  upon  the 
resistances  of  the  primary  of  Motor  II  and  of  the  secondary  of 
Motor  I,  are  obtained  by  dividing  rz  and  rs  respectively  by  slf 

A  diagram  of  equivalent  circuits  is  shown  in  Fig.  120.  At  low 
frequency  a  large  current  may  pass  in  spite  of  the  large  leakage 
field  which  it  produces,  but  this  current,  in  order  to  pass  through 
the  ohmic  resistance  of  the  primary  of  Motor  II  and  the  secondary 
of  Motor  I,  requires  a  magnetic  field  in  the  rotor  of  Motor  I 
whose  magnitude  is  ^i  -r-  st  times  that  which  would  be  required 
for  the  same  current  if  the  frequency  were  ~lm  The  important 
fact  thus  becomes  outstanding  that,  as  the  impressed  frequency 
on  Motor  II  diminishes,  the  effect  of  its  primary  resistance  is 


FIG.  120. — Concatenation:  Equivalent  circuits,  leakage  and  resistance. 

enhanced  to  the  point  that,  at  synchronism  of  Motor  I,  its  effect 
is  equivalent  to  infinite  resistance  or  an  open  circuit  of  the 
primary  of  Motor  II. 

An  inspection  of  this  method  of  connection  of  the  two  motors 
shows  that,  exactly  at  half  synchronous  speed,  the  relative  angu- 
lar velocity  of  the  primary  field  of  Motor  II  in  respect  to  its  rotor 
is  zero.  Therefore,  there  are  no  currents  induced  in  Rotor  II, 
and  the  magnetizing  current  il*  is  the  load  current  of  Rotor  I. 
Motor  II,  therefore,  runs  idle,  as  it  were,  at  half  the  frequency 
impressed  upon  Motor  I,  and  the  torque  of  Motor  I,  since  its 
rotor  current  is  in  quadrature  with  its  rotor  field,  must  vanish. 
The  magnetizing  current  of  the  two  motors,  supplied  through 
the  stator  of  Motor  I,  is  therefore  approximately  twice  the 
magnetizing  current  of  Motor  I  by  itself,  and  equal  to  twice  that 
of  the  group  at  Sj_  =  0. 


156 


INDUCTION  MOTOR 


Now,   consider   the    relation    betwen 
which  there  exists  the  equation 


For 
For 
For 


+1 


*:= 

Si«ii  =  0 

8,   =  0 

*i*n  =  " 


and    sx,    between 
(143) 


-8 


+1 


FIG.  121.  —  The  slip  of  the  second  motor  as  a  function  of  the  slip  of  the  first 
motor  in  concatenation. 
2sj  —  1 


These  relations  are  shown  in  diagram  Fig.  121,  which  also  shows 
the  slip  sn  as  a  function  of  st.  It  is  noticed  that  sn  =  <»  for 
Sj.  =  0. 

We  are  now  prepared  to  examine  more  closely  into  the  relation 


METHODS  OF  SPEED  CONTROL 


157 


of  the  fields,  currents,  and  e.m.fs.  existing  in  the  concatenated 
circuits.  Let  F*,  Fig.  122,  be  the  resultant  magnetic  field  in  the 
secondary  of  Motor  II.  It  induces  an  e.m.f.  e*  according  to 
equation  (138). 

e4  =  2.12(sisn)~iZ4F410-8  volts  (138) 

This  e.m.f.  produces  a  current 

t4  =  e,  +  r,  (144) 


FIG.  122. — Concatenation:  The    flux   diagram.     Leakage  and  resistance  taken 

into  account. 

whose  m.m.f.  is  in  quadrature  with  F4.  The  leakage  field,  /4  = 
ab,  is  in  phase  with  this  m.m.f.  The  flux  $4  =  be  completes  the 
triangle  of  the  magnetic  fluxes  in  which  Oc  =  $3,  the  flux  which 
is  proportional  to  the  primary  m.m.f.  of  Motor  II. 

The  primary  resultant  and  actually  existing  flux  of  Motor  II 
is  Od  =  Fs. 

In  order  to  take  account  of  the  primary  resistance  of  Motor 
II,  we  represent  it  by  that  magnetic  field  which,  at  the  same 


158  INDUCTION  MOTOR 

frequency  as  Fs,  would  induce  the  same  ohmic  drop  z>3.     As 
the  frequency  of  F3  is  (~i  —  ~2)  we  have 

63  =  2.12(si)~iZ3F310-8  (145) 

Therefore 

itfz  :  e3  ::/3r  :F3 


K  (146) 

i 

where  K  =  1  ^-  2.12~iz310-8  (147) 

Hence,  /3r  may  be  represented  by  a  vector  proportional  to  the 
ohmic  drop  divided  by  the  slip  Sj.  It  is  now  clear  that  with 
si  =  -0,  fsr  becomes  infinite,  which  is  the  equivalent  of  a  com- 
pletely open  circuit  of  the  primary  of  Motor  II. 

The  impressed  e.m.f .  of  Motor  II  is  generated  by  the  secondary 
of  Motor  I.  The  relation  of  its  phase  to  its  current  is  determined 
by  the  consideration  that  its  current  is  the  same  as  the  current 
in  the  primary  of  Motor  II  and  its  phase  is  the  same  as  that  of  the 
voltage  impressed  on  the  primary  of  Motor  II  relative  to  its 
current.  The  current  in  circuits  II  and  III  being  the  same,  the 
leakage  fields  and  resistance  drop  fields  must  also  be  the  same. 
Hence 

Og  =  gh  =  //  =  fc    and 
bd  =  di  =  f3    =  fz 

The  actual  resultant  magnetic  field  in  the  secondary  of  Motor 
I  is  now  represented  by  hi  =  F2. 

The  fictitious  flux  3>2  corresponding  to  the  secondary  m.m.f. 
of  Motor  I  is  represented  by  Oc  =  im,  while  hm  =  $1,  the  pri- 
mary fictitious  flux  proportional  to  t\.  The  primary  leakage 
flux  is  ei  =  /i,  and  hi  =  F\  is  the  primary  resultant  flux  generat- 
ing a  counter  e.m.f. 

61  =  2.12— iZi^ilO-8  volts  (49) 

whose  phase  is  in  quadrature  with  ii  and  $1  or  hm. 

An  examination  of  the  diagram  shows  at  a  glance  the  vectorial 
composition  of  the  secondary  flux  ^4  with  the  leakage  fluxes 
/4,  /a,  /2  and /i,  and  the  resistance  drop  fluxes /3r  and/2r,  into  the 
resultant  primary  flux  FI.  It  is  most  interesting  and  instruc- 


METHODS  OF  SPEED  CONTROL  159 

tive  to  note  that  this  composition  can  take  place  actually  in  the 
same  motor  in  the  case  of  internal  concatenation,  which  can  most 
readily  be  realized  by  two  windings  with  numbers  of  poles  in  the 
ratio  of  2  :  1,  as  in  this  case  the  windings  are  mutually  inde- 
pendent in  respect  of  mutual  induction.  A  similar  case  of  great 
theoretical  interest  is  the  case  of  a  poly-phase  motor  with  a  single- 
phase  secondary  which  will  be  discussed  in  Chap.  XII  B. 

The  diagram  of  the  composition  of  fluxes,  Fig.  122,  neglects 
as  usual  the  primary  resistance  of  Motor  I,  which  can  be  taken 
into  consideration  by  the  simple  graphical  correction  given  in 
Chap.  Ill,  if  such  correction  should  prove  desirable. 

It  is  not  necessary,  in  order  to  determine  a  number  of  points 
for  different  speeds  to  develop  more  than  the  flux  polygon 
0-a-b-d-i-e-h.  The  leakage  flux  ab  =  /4  is  proportional  to  the 
secondary  current  of  Motor  II.  The  ohmic  drop  ur^  is  propor- 
tional to  the  flux  Ft  multiplied  by  the  secondary  frequency  of 
Motor  II,  viz.,  (~i  -  2~2).  Therefore,  equation  (138) 

e4  =  z>4  =  2.12(sisn)~iZ4/<T410-8  volts          (148) 

t4    =    #4$4  (149) 

/4    =    (04    - 


.'.t|      =K4^J  (150) 

From  (148)  follows: 


#4/4 


-  1)2.12' 

(152) 


Thus  siSu  being  known  from  (152),  we  obtain 


The  procedure  is  now  as  follows  :  Assume  F*  and  Si  ;  calculate 
from  which  we  obtain  /4;  determine  FA  in  the  usual  manner, 


eb  being  equal  to  (1  --  )$4.     The  direction  and  magnitude  of 

t>3 

the  primary  current  of  Motor  II  thus  being  known  from  /3, 
determine  the  resistance  drop  fields  Og  =  /3'  and  gh  =  /2',  thus 


160  INDUCTION  MOTOR 

obtaining  F2.  The  leakage  field  /2  =  di,  and  the  point  k  are 
then  determined  as  before,  ik  =  (1 )<£2,  and  hi  -£•  hk  =  vi, 

obtaining  FI  =  hi,  in  quadrature  with  which  we  find  e\.  The 
angle  of  lag  is  \f/i,  between  $1  and  e\.  The  construction  of  this 
diagram  has  been  carried  out  for  a  number  of  values  of  the  slip 
«i  and  the  results  have  been  plotted  in  Figs.  123,  124,  125,  126, 
and  127. 

The  torque  of  Motor  II  is  proportional  to  the  vector  product 
of  u  and  F4,  it  is  therefore  represented  by  the  vector  product  of 


FIG.  123. — Concatenation:  Flux  diagram  motor  II  as  generator.     Leakage  and 
resistance  taken  into  account. 

F4  and  /4.  The  torque  of  Motor  I  is  proportional  to  the  product 
of  the  quadrature  component  of  i2t  viz.,  the  component  md  of  the 
leakage  flux  /2  into  the  field  F2.  This  calculation  has  been 
carried  out  and  the  values  of  the  two  torques  have  been  plotted 
in  Figs.  125  and  126. 

It  has  been  customary,  in  popular  theories  of  concatenation,1  to 
assume  that  the  torque  of  Motor  I  follows,  through  the  entire 
range  of  slip  from  sI  =  I  to  sI  =  0,  the  torque  curve  of  an  ordi- 

1  See  F.  EICHBERG,  Zeitschriftfiir  Elektrotechnik,  Vienna,  December,  1898. 
Also  B.  G.  LAMME,  Electric  Journal,  Pittsburgh,  September,  1915. 


METHODS  OF  SPEED  CONTROL 


161 


With  Resistance 


Fio.   124. — The  fluxes  of  concatenated  motors.     Leakage  and  resistance  taken 
into  account  motor  II  acting  as  generator. 


2000 


2000 


FIG.  125. — The  torque  curves  of  two  concatenated  motors.     T\  is  the  torque  of 
motor  I.     Tz  is  the  torque  of  motor  II.     T\  +  Tz  is  the  resultant  torque. 


11 


162 


INDUCTION  MOTOR 


nary  induction  motor  at  constant  potential,  while  the  torque 
curve  of  Motor  II  is  assumed  to  be  that  of  a  constant  potential 
motor  whose  synchronism  is  reached  at  half  the  synchronous 


2000 

G'enert'r 


Indicated  by 


DIAGRAM  OF  FLUXES 
MOTOR 


tg  P=^ 


FIG.  126. — Concatenation:  The  torques  and  the  loci  of  the  primary  current' 
Leakage  and  resistance  taken  into  account. 

speed  of  Motor  I  operating  by  itself,  in  Fig.  128.     It  is  clear 
from  our  analysis  and  also  from  the  consideration  of  the  fact  that 


METHODS  OF  SPEED  CONTROL 


163 


the  torque  of  Motor  I  for  sl  =  0.5  must  be  zero,  that  this  plausible 
explanation  is  incorrect,  and  misleading  in  regard  to  the  physical 
aspects  of  this  problem. 


3  V  r3=  0 


=  .21 


10 


FIG.  127. — Concatenation:  The  primary  current  locus  of  two  concatenated 
motors  with  leakage  and  resistance  taken  into  account. 


FIG.  128. — Convential  but  incorrect  method  of  representing  the  torque  of 
two  concatenated  motors.  (3)  Torque  of  one  motor  operating  at  double  slip. 
(2)  Torque  of  other  motor  operating  at  normal  slip.  (1)  Torque  of  the  con- 
catenated group. 

It  is  necessary  to  consider  carefully  the  effect  of  the  second 
motor  on  the  characteristics  of  the  group.  Above  half  synchron- 


164 


INDUCTION  MOTOR 


ous  speed,  Sj  =  0.5,  Motor  II  runs  above  its  synchronous  speed 
relative  to  its  supply  frequency  ^i  —  ^2,  and  therefore  acts 
throughout  its  range  to  s,  =  0  as  an  induction  generator.  Yet 
the  effect  of  the  resistances  r2  and  r3  consists  in  changing  over 
the  torque  of  Motor  I  between  sr  =  0.3  and  st  =  0,  from  a  gen- 
erator torque  into  a  motor  torque.  This  point  is  so  important 
that  we  give  in  Fig.  123  and  124  a  complete  polar  diagram  for  this 
condition,  which  should  now  be  self-explanatory. 


FIG.  129.  —  Concatenation:  Resistance  only  and  no  leakage. 

as  motor. 


Motor  II  acting 


From  Fig.  125  it  is  evident  that  the  torques  of  both  motors  are 
motor  torques  up  to  st  =  0.5,  while  from  st  =  0.5  to  sx  =  0.3 
both  torques  are  generator  torques.  Between  sI  =  0.3  and  sz  = 
0  Motor  I  is  a  motor,  while  Motor  II  remains  a  generator.  At 
negative  slips  the  group  acts  individually  and  collectively  as  a 
generating  unit.  Figure  127  shows  the  polar  diagram  of  the 
primary  current  of  Motor  I.  The  locus  of  this  curent  is  a  curve 
of  the  fourth  power.  The  slip  sr  is  marked  everywhere  and  it  is 
interesting  to  note  how  the  current  from  standstill  gradually 
diminishes  to  half-synchronous  speed,  as  in  a  single  ordinary 
induction  motor.  The  magnetizing  current  at  half  -synchronous 
speed  is  almost  double  that  of  a  single  motor.  Above  this  speed 
the  group  acts  like  an  induction  generator  until,  as  a  result  of  the 
effect  of  the  secondary  resistance  of  Motor  I  and  the  primary 


METHODS  OF  SPEED  CONTROL 


165 


resistance  of  Motor  II  above  sI  =  0.21  the  group  becomes  a 
motor,  and  the  primary  current,  while  at  first  increasing,  is 
gradually  choked  off  by  these  resistances  until  &t  sI  =  0  their 


FIG.   130. — Concatenation:  Resistance  only  and  no  leakage.     Motor  II  acting 

as  generator. 


12  r2    _  12.5    X.I 


Sf 


-.15 
=  8.35 


FIG.    131. — Concatenation:    Resistance    only,       FIG.  132.  —  Concatenation: 
no  leakage.  Resistance  only,  no  leakage. 

effect  becomes  equivalent  to  an  open  circuit  of  the  primary  of 
Motor  II  and  the  primary  current  of  Motor  I  drops  to  the  value 
of  its  magnetizing  current. 

To  bring  out  this  very  interesting  but  somewhat  involved 


166 


INDUCTION  MOTOR 


process  more  clearly,  we  shall  consider  two  specific  cases.  First, 
two  equal  motors  in  concatenation,  without  leakage,  but  with 
resistance  r2  and  r3;  and,  secondly,  two  equal  motors  with  leakage 
but  without  resistance  r2  and  r3.  Neither  case  corresponds  to  the 


POLAR   DIAGRAM 

CONCATENATION  RESISTANCE  ONLY 


£,=.2 


FIG.  133. — Concatenation:  Resistance    only,    no    leakage.     Locus    of    primary 

current. 

case  treated  fully  in  this  chapter,  but  as  " boundary"  solutions, 
they  afford  a  good  insight  into  the  operation  of  concatenation. 

We  assume  r2  =  r3  and  draw  F*.  (Figs.  129  and  130.)  Deter- 
mine $4  =  bd  as  before,  and  draw  ad  =  <J>2  =  $3.  The  resistance 
drop  field  ft  =  ft  is  shown  as  be  =  2/J.  Join  ac  =  Fz  =  Fs  and 


METHODS  OF  SPEED  CONTROL 


167 


Qa  =  FI  =  Fa.  The  primary  voltage  e\  is  in  quadrature  with 
FI,  while  Od  =  $1  represents  the  phase  and  magnitude  (including 
a  proportionality  factor)  of  the  primary  current  i\.  Angle  bad  = 
0,  and  tgO  =  $4  -r-  F4  =  s^.  The  resistance  drop  field  be  — 
2/J  is  also  equal  to 

V«T« 

(154) 


where  X  is  a  proportionality  factor  readily  obtained  as  before. 

Figure  133  shows  the  polar  diagram  of  the  primary  current  for 
different  slips.  Its  relation  to  the  fourth  degree  curve  of  Fig. 
127  is  striking.  Figures  131  and  132  are  auxiliary  diagrams  used 
in  the  development  of  the  diagram.  As  in  the  ordinary  induc- 
tion motor  without  leakage,  the  branches  of  the  curve  are 
asymptotic  to  the  ordinate  axis. 


CONCATENATION  OF  Two  EQUAL  MOTORS 

Resistance  Only 

No  Leakage 


«I 

tgP 

COS  \f/i 

ii 

2.000 

7.500 

0.964 

143.5 

1.000 

2.500 

0.785 

65.5 

0.900 

2.000 

0.706 

59.0 

0.800 

1.500 

0.620 

54.0 

0.700 

1.000 

0.480 

50.0 

0.600 

0.500 

0.300 

48.5 

0.500 

0.000 

0.093 

49.0 

0.400 

0.500 

-0.105 

56.0 

0.300 

-1.000 

0.200 

72.5 

0.250 

-1.250 

-0.166 

87.0 

0.200 

-1.500 

-0.895 

106.0 

0.175 

-1.630 

0.340 

106.0 

0.150 

-1.750 

0.506 

94.6 

0.125 

-1.877 

0.690 

74.5 

0.100 

-2.000 

0.700 

53.5 

0.050 

-2.250 

0.530 

30.8 

0.000 

0.000 

0.000 

25.0 

-0.100 

-3.000 

-0.535 

30.6 

-0.200 

-3.500 

-0.760 

44.0 

-0.300 

-4.000 

-0.820 

53.0 

-0.600 

-5.500 

-0.920 

78.5 

-1.000 

-7.500 

-0.960 

110.5 

168 


INDUCTION  MOTOR 


Figure  134  shows  the  current  curves  in  polar  coordinates  for 
this  limiting  case,  and  Fig.  135  the  equivalent  electric  circuits 
which  are  always  instructive. 


FIG.  134. — Primary   current  locus.     Concatenation.     Resistance  only.     Num- 
erals represent  slip  sj. 

Secondly,  we  shall  now  consider  the  action  of  the  group  of  the 
two  motors  if  the  resistances  r2  and  r3  are  neglected,  the  leakage, 
however,  being  fully  taken  into  account.  This  is  a  very  im- 
portant case  as  it  will  be  shown  that,  for  a  range  of  slip  from 


METHODS  OF  SPEED  CONTROL 


169 


§!  =  1  to  sx  =  0.4,  which  is  the  important  range  for  practical 
purposes,  this  diagram  becomes  very  simple,  the  locus  of  the 
primary  current  being  again  a  circle. 

We  begin  again  with  the  secondary  flux,  F-*  of  Motor  II.     It 
induces  an  e.m.f. 

e4  =  2.12(sIsII)~iZ4/'74lO-8  volts  (155) 


•WV— AM, 


FIG.  135. — Concatenation:  Equivalent  circuits  resistance  only;  no  leakage. 

This  e.m.f.  produces  a  current 

*4  =  -  (156) 


This    current    produces   a   leakage    field  /4  =  ab  =  (vz  — 

<l>4  and  F4  are  combined  into  $3  and  fa  =  be  is  the  leakage  field  in 

magnitude  and  phase  of  the  primary  m.m.f.  of  Motor  II. 


— UL 


FIG.  136. — Concatenation:  Equivalent    circuits    leakage    reactance    only;    no 

resistance. 

As  our  premise  was  the  assumption  of  a  negligible  resistance 
of  the  windings  between  the  two  motors,  see  Ffg.  134  which 
assumption  we  know  to  be  permissible  only  if  the  slip  sr  is 
not  less  than  60  per  cent,  we  can  directly  combine  the  field  F^ 
which  is  the  resultant  primary  magnetic  field  of  Motor  II,  with 
its  secondary  fictitious  flux  3>i,  and  we  obtain  the  fictitious 
primary  field  whose  leakage  field  is  /i  =  mu. 


170 


INDUCTION  MOTOR 


The  diagram  is  developed  exactly  as  the  flux  diagram  of  the 
induction  motor,  and  every  step  should  be  carefully  thought 
over  by  the  reader.  (Fig.  137.) 

An  examination-  of  this  geometrical  figure  now  shows  the 
interesting  and  remarkable  fact  that,  with  constant  impressed 
voltage  on  Motor  I,  neglecting  its  primary  resistance  which  is 
permissible,  the  point  of  intersection  /  remains  fixed  for  all 
speeds  of  Rotor  II,  and  the  primary  flux  3>i  of  Motor  I  moves 
under  this  condition  upon  the  periphery  of  the  circle  around  D 
as  center 

This  can  readily  be  proved  as  follows : 


FIG.  137. — Flux   diagram   of   two   motors   in   concatenation.     No   primary 
and  secondary  resistance  in  motor  I,  and  no  primaary  resistance  in  motor  II. 

Draw  as  and  sd  and  obtain  an  expression  for  sf  =  sd  +  df. 


(157) 


ed:df::F2:2F2 

.'.  df  =  2ed 
nd  =  $2(^1  —  1) 
dm  =  $2(^2  —  1) 


mk 


d    =    C|>2   -f    hi 

111  =  nd  -f  dm  +  mk 


.  Cl    = 

cl  = 


(158) 
(159) 


METHODS  OF  SPEED  CONTROL  171 

X  =  [fli  +  w«  -  -1  (160) 

v\j 


i:kl:  :d:cl 


(161) 

(162) 

if  =  kl  —  2ed 

ty  =  *4(2£-l)  (163) 

a6  =  *4(02  —  1) 


ar:ds::  —  : 


.:  ds  =  ZQ^VM  -  1)  (164) 

df  =  2ed  =  2*4(1  -  p)    .  (165) 

.-.*/=  2*4  »i0s  -  (166) 


This  equation  shows  that  s/  is  proportional  to  *4  and  it  is  there- 
fore a  measure  of  $4. 

0/:ce::2:l 
.-.0/  =  2ce 
ce:ck::  ei  :  kl 

F1     .vi  . 
ce:  —  :  :  <J>4-  :  *4 

Vi  A 


~  (167) 

A 


But  Of  =2ce 


/.  O/  =  2^  (168) 

A 


172  INDUCTION  MOTOR 

From  similar  triangles: 

Qf:D:i8f:if 


%->* 


Of      2aifaX  - 
D 


(169) 


Without 
Resistance 


FIG.   138.  —  The  fluxes  of  concatenated  motors.     Leakage  only  (not  considering 
resistance).     The  group  acting  as  generator. 

The  Fig.  137  is  drawn  for  vi  =  v2  =  1.1,  and  FI  =  49  and 
therefore 

X  =  vi  +  vz  -- 

01 

X  =  1.29 


=  76 
J'=  1.016 

D  =  74.8 


METHODS  OF  SPEED  CONTROL 


173 


Figure  138  shows  an  auxiliary  diagram  in  which  both  motors 
are  operating  as  induction  generators. 

Figure  139  shows  a  comparison  of  a  single  motor,  as  one  of 
the  concatenated  group,  with  the  concatentated  group,  with 
stray  coefficients  vi  =  v2  =  1.04,  or  a  =  .082. 


FIG.  139. — Concatenation:  The    influence    of   leakage.     Approximate    primary 

current  loci  of  group  and  of  single  motor. 

Vl=vz  =  1.04 

Figure  140  shows  the  same  groups  of  motors  with  stray  coeffi- 
cients Vi  =  vz  =  1.1,  or  <r  =  .21. 

The  greater  the  leakage,  the  less  advantageous  appears  to  be 
the  grouping  of  the  motors  in  concatenation  as  their  joint 
capacity  and  maximum  output  are  enormously  reduced. 


/Concatenated\     Single  Motor 
/  Group       VX 

**5 


FIG.  140. — Concatenation:  The    influence    of    leakage.     Approximate    primary 

current  loci  of  group  and  of  single  motor. 

»i   =  vi  =  1.1 


B.  THE  POLY-PHASE  MOTOR  WITH  SINGLE-PHASE  SECONDARY 

H.  Goerges  discovered  in  the  early  nineties  that  in  an  induc- 
tion motor  of  the  slip-ring  type  if,  after  starting,  one  brush  is 
lifted  on  the  rotor,  the  rotor  will  accelerate  until  it  has  reached 
half  speed,  at  which  speed  it  will  run  idle.  This  is  a  very  curious 
and  interesting  result. 


174 


INDUCTION  MOTOR 


In  view  of  the  fact  that  heavy  currents  must  be  induced  in  a 
uni-axial  rotor  winding  at  half  synchronous  speed  by  the  re- 
sultant rotating  field  in  respect  to  which  the  rotor  has  a  slip  of 
50  per  cent,  it  is  apparent  that  at  half  speed  such  a  poly-phase 
motor  with  single-phase  secondary  must  have  a  larger  mag- 
netizing current  by  the  amount  necessary  to  compensate  for  the 
rotor  currents. 

If  we  resolve  the  single-phase  m.m.f.  belt  which  exists  in 
the  rotor  into  two  m.m.f.  belts  of  half  the  amplitude,  rotating 
in  opposite  directions,  then  we  obtain  the  following: 

Secondary  frequency  ^i  —  ^2,  to  which  corresponds  clock- 
wise an  angular  velocity  in  space  co2  -}-  wi  —  co2  =  coi,  if  the 

KW 


FIG.  141. — Torque  curve  of   poly-phase   motor  with   uni-axial  rotor  winding. 
(From  Arnold,  vol  V,  part  1.) 

rotor  revolves  clockwise.  Counter-clockwise  there  appears 
oo2  —  (coi  —  co2)  =  —  (coi  —  2co2)  as  the  angular  velocity  in  space 
of  the  second  component  of  the  resolved  single-phase  m.m.f. 
belt. 

The  first  component  rotating  in  space  with  angular  velocity 
coi  may  be  combined  with  the  impressed  m.m.f.  belt  of  the 
primary. 

The  second  component  rotating  in  space  with  angular  velocity 
—  (coi  —  2o>2)  may  be  looked  upon  as  the  impressed  m.m.f.  belt 
of  a  motor  in  which  the  rotor  is  the  primary  and  in  which  the 
stator,  short-circuited  as  it  were  through  its  supply  circuit, 
forms  the  secondary. 

Thus  we  obtain  the  identical  conditions  of  internal  concatena- 


METHODS  OF  SPEED  CONTROL  175 

tion,  and  the  general  theory  of  concatenation  may  with  propriety 
be  applied  to  this  subject. 

For  a  similar  statement  of  the  theory  of  this  phenomenon,  see 
especially  E.  Arnold,  "Les  Machines  d'Induction,"  Paris,  Ch. 
Delagrave,  1912,  p.  184,  where  an  experimental  diagram  is  given 
of  the  torque  of  such  a  motor  as  a  function  of  the  slip,  see  Fig. 
141,  which  is  like  Fig.  125  in  the  Chap.  XII.  A.  Mr.  B.  G.  Lamme 
in  "Electrical  Engineering  Papers,"  p.  519,  has  given  the  same 
general  explanation.  We  differ,  however,  from  him  as  to  the 
propriety  of  deriving  the  resultant  torque  of  such  a  motor  from 
two  constant-potential  torque  curves  for  different  slips.  It  is 
evident  that,  at  half  speed,  the  currents  in  the  rotor  are  almost 
totally  watt-less;  therefore,  there  can  be  no  torque  at  that  speed. 
The  method  reprinted  by  F.  Eichberg  in  his  "Collected  Papers," 
p.  82,  is  open  to  the  same  criticism.  The  two  fictitious  motors  of 
this  combination  do  not  operate,  either  under  constant  potential, 
or  at  all  similarly  to  the  standard  induction  motors  with  short- 
circuited  rotor  as  it  must  have  become  clear  in  discussing  con- 
catenation in  the  previous  chapter. 

The  reflection  of  low-frequency  currents  into  the  primary  of 
the  motor  through  induction  from  the  second  hypothetical 
motor  whose  secondary  is  the  main  primary  circuit,  short-cir- 
cuited through  the  supply  circuit  as  such,  leads  to  disturbances 
in  the  supply  circuit  which  make  a  practical  application  of  this 
ingenious  scheme  most  undesirable.  Reference  to  this  is  found 
also  in  Arnold's  book,  loc.  cit.,  p.  185. 


CHAPTER  XIII 
METHODS  OF  SPEED  CONTROL  (Continued') 

C.    CONCATENATION    OF   AN    INDUCTION    MOTOR    WITH    THE 

COMMUTATOR  TYPES  FOR  THE  INDUCTION  OF  A 

SLIP  FREQUENCY  E.M.F. 

All  the  methods  devised  for  obtaining  a  change  in  speed  by 
means  of  concatenation  depend  upon  the  generation  of  currents 
of  slip  frequency  properly  injected  into  the  rotor  of  the  motor 
which  is  to  be  controlled. 

(a)  Poly-phase  commutator  motors  of  the  series  or  shunt  or 
compound  type  may  be  mounted  on  the  same  shaft  with  the 
induction  motor  which  is  to  be  regulated.  The  series  commutator 
motor  may  have  salient  poles  to  obtain  increased  stability  like  the 


FIG.  142. — "Kraemer   System,"    Induction   motor   mechanically   concatenated 
to  poly-phase  commutator  induction  motor. 

machines  of  F.  Lydall,  A.  Scherbius,  and  Miles  Walker.     This 
system  is  called  the  "Kraemer  System."     (Fig.  142.) 

(6)  Instead  of  combining  the  regulating  machine  with  the  main 
motor  into  one  mechanical  unit,  they  may  be  separated  as  was 
done  by  A.  Scherbius.  (Fig.  143.) 

(c)  A  rotary  converter  may  be  mounted  on  the  same  shaft 
with  the  induction  motor,  the  combination  operating  at  half 
speed  for  equal  numbers  of  poles,  and  direct  current  power  be- 
coming available.     Method  proposed  by  J.  L.  la  Cour.     (Fig.  144.) 

(d)  The  rotary  converter  may  be  separated  from  the  shaft 
of  the  induction  motor  and  a  d.c.  machine  may  be  controlled 
on  the  same  shaft  as  the  induction  motor  by  this  rotary  converter. 
(Fig.  145.) 

176 


METHODS  OF  SPEED  CONTROL 


111 


(e)  The  d.c.  machine  on  the  shaft  of  the  induction  motor  may 
be  replaced  by  an  independently  running  set  of  d.c.  motor  and 
induction  generator.  (Fig.  146.) 


FIG.  143. — "Scherbius  System,"  Induction  motor  electrically  concatenated 
through  poly-phase  commutator  motor  which  delivers  current  back  into  the 
line. 


D.C. Excitation 


l.M. 


D.C. 


FIG.  144. — Combination  of  in- 
duction motor  with  rotary  con- 
verter. (J.  L.  Lacour). 


FIG.  145. — Speed  regulation  of  induction 
motor  by  means  of  rotary  converter  and 
direct  connected  D.  C.  machine. 


(/)  Another   method   was   devised   by 

Ruedenberg  and  it  is  shown  in  Fig.  147. 
12 


A.   Heyland   and   R. 


178 


INDUCTION  MOTOR 


FIG.  146. — Speed  regulation  of  induction  motor  by  means  of  rotary  converter  and 
separately  driven  direct  current  induction  motor  generator  set. 


FIG.  147. — Combination  of  induction  motor  with  commutator  motor  and  small 
induction  motor.     (Heyland-Ruedenberg). 


FIG.  148. — Speed  regulation  of  induction  motor  by  means  of  synchronous  motor 
direct  connected  and  fed  from  frequency  changer  driven  by  small  synchronous 
motor.  (Brown,  Boveri  system). 


0 


(Facing  page  178) 


METHODS  OF  SPEED  CONTROL  179 

(g)  The  latest  and  perhaps  principally  the  simplest  method 
is  described  in  the  German  patent  No.  264,673,  July  24,  1910, 
taken  out  by  Brown  Boveri  &  Company.  It  is  shown  in  Fig. 
148,  where  S.  M.  represents  a  synchronous  motor  mounted  on 
the  shaft  of  the  induction  motor,  F  is  a  frequency  changer  of  the 
type  first  suggested  by  B.  G.  Lamme,  and  S  is  again  a  small 
synchronous  motor. 

D.  CHANGE  OF  SPEED  BY  CHANGING  THE  NUMBER  OF  POLES 

A  great  many  windings  have  been  devised  to  obtain  different 
numbers  of  poles  with  one  winding.  Two  or  more  entirely 
separate  windings  wound  for  different  numbers  of  poles  have 
been  used  in  the  same  slots,  either  with  corresponding  rotor 
windings,  or  with  a  squirrel-cage  rotor. 

By  utilizing  a  winding  of  a  fractional  pitch,  it  is  possible  to 
split  it  in  such  a  manner  as  to  arrange  it  for  two  numbers  of 
poles,  and  even  for  four  numbers.  These  windings  are  ingenious 
but  intricate  and  their  field  of  application  is  limited. 

It  is  important  to  bear  in  mind  in  designing  such  machines 
that  the  leakage  factor  of  a  winding  of  twice  the  number  of 
poles  is  very  nearly  twice  as  great.  Great  attention  must  also 
be  paid  to  the  magnetizing  current,  otherwise  it  might  become 
excessive  in  view  of  the  reduction  of  the  active  conductors  per 
pole.  The  subject  is  too  broad  to  be  treated  here  at  length, 
but  I  shall  indicate  one  or  two  methods  to  outline  the  general 
principle  and  describe  one  of  the  most  effective  methods  used 
to  obtain  this  change  in  the  number  of  poles. 

1.  The  oldest  method  consists  in  the  use  of  two  or  more 
sets   of  stator  coils,  the  pitch  of  which  is  such  as  to  give  a 
number  of  windings  with  a  different  number  of  poles1  for  each. 
Such  a  scheme  is  impracticable  as  it  is  wasteful  of  space  and 
material. 

2.  A  single  winding  can  be  used  which,  due  to  different  lap 
caused  by  the  coil-pitch  being  less  than  the  pole-pitch,  may  be  so 
connected  as  to  produce  two  different  numbers  of  poles. 

3.  The  ingenious  winding  of  Alexanderson,2  by  which  four 
numbers  of  poles,  as  6,  8,  12,  and  24,  may  be  obtained  with  a 

1  See,  for  instance,  B.  G.  LAMME,  U.  S.  Patent  No.  660,909,  Oct.  30,  1900. 

2  E.   F.   W.  ALEXANDERSON.   U.  S.   Patents  Nos.   841,609  and  841,610, 
Jan.  15,  1907. 


180 


INDUCTION  MOTOR 


winding  the  individual  coils  of  which  are  all  alike  but  the  con- 
nections of  which,  by  means  of  30  leads  brought  out  from  the 
motor,  are  connected  through  a  drum  controller.  Six,  eight,  and 
twelve  poles  are  obtained  by  the  use  of  only  12  leads.  The 


.  SfcEStJi  i 

<^rr;r 

Vvtf- -\V/'-}-|— is 

iU~ 


3l&i 


o\A-t?\^fr 

*)ter  i 

•^^ri11 


FIG.  149.  —  E.  F.  W.  Alexanderson's  stator  winding  for  three  or  four  different 
numbers   of   poles.     (From   U.    S.   patent   No.   841,609,    Jan.    15,    1907.) 


accompanying  winding  diagrams,  taken  from  Alexanderson's 
patent,  show  clearly  the  arrangement  of  the  circuits.  (Figs.  149 
and  150.) 

The  Oerlikon  Company1  has  built  motors  of  this  kind  with  a 

JSee  E.  T.  Z.,  1914,  No.  31. 


METHODS  OF  SPEED  CONTROL 


181 


double  rotor,  one  inside  the  other,  and  each  motor  arranged  for 
pole-changing.  The  motors  are  so  arranged  that  one  motor 
drives  the  other  and  with  two  sets  of  poles  on  the  main  motor 


10 


20 


10  20 


24  Poles 


24  Poles 

FIG.  150. — E.  F.  W.  Alexanderson's  stator  winding  for  three  or  four  different 
numbers  of  poles.  Arrows  show  direction  of  currents  in  the  different  phases  per 
pole.  (From  U.  S.  Patent  No.  841,609,  Jan  15,  1907). 

and  six  sets  of  poles  on  the  auxiliary  motor,  it  can  be  seen  that 
226  +  2  =  26  different  speeds  can  be  obtained. 


CHAPTER  XIV 

TYPES  OF  VARIABLE  SPEED  POLY-PHASE  COMMUTA- 
TOR MOTORS 

Having  explained  at  great  length  in  previous  chapters  the 
principles  upon  which  is  based  the  speed  control  of  induction 
motors,  we  shall  consider  briefly  three  types  of  motor  which 
embody  these  principles  in  one  unit. 

A.  The  Plain  Shunt  Motor. — This  motor  in  its  simplest  form 
is  represented  by  Fig.  151.  The  voltage  on  the  rotor  brushes 
as  well  as  their  phase  can  be  varied.  The  brushes  can  be  shifted. 


FIG.  151.— Poly-phase  shunt  motor.  Without  transformer  and  requiring 
mechanical  brushshift.  (This  motor  is  identical  with  Heyland's  "Compensated. 
Motor,"  excepting  for  the  shunts  between  commutator  bars,  later  introduced  and 
abandoned.) 

Both  speed  regulation  and  power  factor  regulation  may  be  ob- 
tained. This  is  the  Goerges  motor  in  its  simplest  form.  Inter- 
posing of  transformers  to  give  different  voltages  and  phases 
suggests  itself  and  innumerable  combinations  can  be  effected. 
(Fig.  152.) 

182 


VARIABLE  SPEED  POLY-PHASE  COMMUTATOR  MOTORS      183 

B.-J.  L.  la  Cour  has  added  a  second  stator  winding  in  order  to 
obviate  the  mechanical  shifting  of  the  brushes  which  is  detri- 
mental, as  it  incurs  higher  harmonics  which  are  very  serious  in 


Stator 

FIG.   152. — Polyphase    shunt    motor    with    transformer,    requiring    mechanical 

brush  shift. 

connection    with    commutation.     Separate    regulation    of    this 
winding  through  a  transformer  is  necessary.     (Fig.  153.) 


FIG.  153. — Motor  of  J.   L.   Lacour   with   additional   stator   winding   to   avoid 
mechanical  brush  shift. 

C.-M.  Osnos,  E.  T.  Z.,  Dec.  11, 1902,  describes  a  motor  in  which 
the  rotor  is  used  as  the  primary  and  the  stator  as  the  secondary. 


184 


INDUCTION  MOTOR 


The  rotor  is  built  with  a  commutator  on  one  side  and  slip  rings 
on  the  other.  The  groups  of  coils  between  the  brushes  carry 
currents  of  the  slip  frequency,  therefore,  to  the  stationary,  form- 
ing the  secondary,  then  the  combination  lends  itself  directly  to 
speed  regulation,  the  proper  voltage  necessary  being  obtained 
by  a  pair  of  brushes  for  each  phase  of  the  secondary.  (Fig.  154.) 
D.-H.  K.  Schrage,  in  U.  S.  Patent  No.  1,079,994  Dec.  2, 
1913,  describes  a  similar  motor  in  which  he  has  added  another 


Rotor,  or  Primary 

Winding  Acting 

as  Regulating 

Winding  also^  \     \  Stator  or 

Secondary 
'Winding 


FIG.  154. — The    variable    speed    commutator    induction    motor    of    Osnos. 

winding,  called  regulating  winding,  which  is  connected  to  the 
commutator  while  the  slip  rings  are  connected  to  an  independent 
primary  winding.  Through  this  modification  of  the  Osnos 
motor  greater  freedom  in  the  choice  of  secondary  voltages  is 
obtained.  The  drawback  of  these  arrangements  seems  to  be 
that,  while  in  the  shunt-motor  type  with  stationary  primary 
the  e.m.f.  short-circuited  under  the  brushes  varies  proportion- 
ally to  the  slip,  and  disappears  at  synchronism,  in  the  Osnos 
and  Schrage  motors  it  remains  constant  over  the  entire  range 
at  all  speeds.  (Fig.  155.) 


VARIABLE  SPEED  POLY-PHASE  COMMUTATOR  MOTORS      185 


Auxiliary 
Regulating  Winding 


Stator  or 
Secondary  Winding 


Rotor,  or 
Primary  Winding 


Fio.  155. — The  variable  speed  commutator  induction  motor  of  H.  K.  Schrage. 
(U.  S.  Patent  1,979,994,  Dec.  2,  1913.) 


Stator  Winding 

FIG.  156. — Shunt    poly-phase    commutator    motor    with    auxiliary    regulating 
winding  on  stator.     (Inversion  of  Schrage  motor.) 


186  INDUCTION  MOTOR 

The  Schrage  motor  is  in  reality  a  kind  of  frequency  transformer 
like  that  described  in  Chap.  IX,  F. 

E.  If  a  regulating  winding  were  added  to  the  motor  described 
under  A,  but  placed  on  the  stator  instead  of  the  rotor  we 
obtain  the  inversion  of  the  Schrage  motor,  Fig.  156. 

NOTE. — In  connection  with  the  subject  of  commutation  in  these  motors, 
briefly  referred  to  in  this  chapter,  see  the  very  clear  exposition  by  B.  G. 
LAMME,  Journal,  A.  I.  E.  E.,  1920,  "The  Alternating  Current  Commutator 
Motor." 


(Facing  page  187) 


CHAPTER  XV 

METHODS  OF  RAISING  THE  POWER  FACTOR  OF 
INDUCTION  MOTORS 

A.  THE  METHOD  OF  LEBLANC  USING  COMMUTATOR  MACHINES 

FOR  SECONDARY  EXCITATION 

To  the  genius  of  Maurice  Leblanc  we  owe  the  methods  for 
raising  the  power  factor  of  induction  motors  by  introducing 
e.m.fs.  of  proper  phase  and  frequency  into  the  rotor.  In  U.  S. 
Patent  No.  613,204,  Oct.  25,  1898,  he  describes  a  method  of 
using  two  or  three  single-phase  commutator  machines  excited 
with  slip  frequency  in  such  a  manner  that  leading  currents  are 
induced  in  the  rotor.  As  a  matter  of  historical  interest,  there 
is  reproduced  in  facsimile  the  illustration  from  the  patent 
specification.  (Fig.  157.) 

The  theory  of  these  interesting  machines  has  been  treated 
in  Chap.  IX,  in  which  it  was  shown  how  a  secondary  phase  lag 
or  lead  affects  the  primary  current  locus,  and  we  therefore  need 
not  repeat  the  subject. 

B.  THE  USE  OF  A  POLYPHASE  COMMUTATOR  FOR  THE  GENERA- 

TION OF  LEADING  CURRENTS 

As  indicated  in  Chap.  VIII,  B,  a  commutator  machine  without 
excitation,  fed  with  polyphase  currents,  generates  at  a  proper 
speed  an  e.m.f.  which  lags  behind  the  exciting  current.  Thus 
the  arrangement  of  Leblanc  may  be  replaced  by  a  single  poly- 
phase commutator  armature  without  excitation,  operated  only 
at  a  high  enough  speed  above  synchronism  to  obtain  the  effect 
required.  This  device  has  been  used  by  Leblanc,  Scherbius, 
Brown  Boveri  &  Company,  etc.,  and  it  is  as  ingenious  as  it  is 
simple.  Its  theory  has  been  given  above. 

An  ingenious  modification  of  this  device  is  used  by  the  Brown 
Boveri  &  Company.  As  the  stator  is  evidently  needed  solely  to 
close  the  magnetic  circuit,  it  may  be  made  integral  with  the 
rotor,  without  an  air-gap,  and  the  stator — forgetful  of  its  name 
and  connotation — revolves  integrally  with  the  rotor. 

The  saturation  of  the  iron  is  high  so  that,  after  high  currents 
are  reached  as  the  result  of  increasing  load,  the  compensating 

187 


188 


INDUCTION  MOTOR 


No.  613,204. 


Patented  Oct.  25,  1898. 


tNo  Model.) 


M.  HUTIN  &  M.  LEBLANC. 
ALTERNATING  CURRENT  ASYNCHRONOUS  MACHINE. 

(Application  filed  May  4,  1807.) 

4  Sheets— Sheet  3. 


FIG.  157. — The  method  of  Leblanc  for  raising  the  power  factor  of  induction 
motors.     (Facsimile    of    the    American    patent    specification.) 


THE  POWER  FACTOR  OF  INDUCTION  MOTORS 


189 


effect  diminishes  resulting  in  a  polar  diagram  as  indicated  ap- 
proximately in  Fig.  158.  Thus  the  power  factor  is  practically 
constant  and  near  unity  over  a  wide  range. 


With  Leblanc-Scherbius  Rotor 
and  Saturation 


Primary  Current  without  Leblanc  Rotor 
with 


without  Saturation 
with 


FIG.  158. —  The  primary  current  of  the  induction  motor  with  or  without  Leblanc- 
Scherbius  rotor. 

C.  THE  METHOD  OF  LEBLANC  INDUCING  LEADING  CURRENTS 

THROUGH    RAPID    OSCILLATION    OF    AN    ARMATURE    IN    A 

MAGNETIC  FIELD 

Maurice  Leblanc  described  an  ingenious  scheme  for  the  gen- 
eration of  leading  currents  in  U.  S.  Patent  No.  644,554,  Feb. 
27,  1900,  with  the  suggestion  that  it  be  used 
in  the  secondary  of  a  slip-ring  type  of  induc- 
tion motor.  Figure  159  shows  the  principle. 

A  coil  a  —  a  is  suspended  between  the  poles 
N  and  S  where  it  can  swing  freely  in  the  mag- 
netic field  established  in  the  air-gap.  The 
low-frequency  alternating  current  upon  which 
an  e.m.f.  producing  a  leading  current  is  to  be 
impressed  by  the  oscillation  of  this  device, 
called  by  the  inventor  a  ' 'recuperator,"  tra- 
verses the  coil  a  —  a. 

The  stronger  the  magnetic  field  and  the 
lighter  the  frame  of  the  coil  and  the  smaller  the  frequency  of  the 


FIG.  159. — Device 
illustrating  the  prin- 
ciple of  the  Leblanc 
"Recuperator." 


190  INDUCTION  MOTOR 

current  passing  through  the  coil  a  —  a,  the  greater  will  be  the 
effective  e.m.f .  due  to  the  oscillation  causing  a  leading  current  to 
be  induced  in  the  circuit  a  —  a.  If  the  moment  of  inertia  of  the 
coil  is  great,  lagging  current  may  be  induced. 

An  interesting,  but  not  very  practical,  method  of  giving 
concrete  shape  to  this  idea  is  shown  in  Fig.  160,  taken  from  Le- 
blanc's  patent.  A  disc  swings  in  a  strong  magnetic  field  into 
which  current  is  conducted  by  means  of  the  ring  R  and  collected 
at  the  circumference  through  a  mercury-trough  M. 

The  theory  is  interesting  and  it  is  clearly  stated  by  Leblanc 
and  Kapp  (see  later).  The  fundamental  idea  consists  in  making 
a  device,  like  a  coil  spring,  which  through  its  oscillations  induces 
an  e.m.f.  in  the  circuit  from  which  originates  the  forced  frequency. 
The  effect  of  such  a  coil  spring  is  like  the  effect  of  the 
"recuperator." 


FIG.  160. — A  possible  practical  form  suggested  by  Leblanc  for  his  "  Recuperator." 

In  a  general  way  it  is  apparent  that  we  can  devise  two  types 
of  dynamic  systems.  One,  in  which  a  heavy  mass  is  set  into 
oscillation;  another,  in  which  a  very  small  mass  attached  to  a 
powerful  spring,  oscillates. 

It  is  shown  in  the  theory  of  dynamics  that  the  velocity  of  the 
oscillating  system  is  a  maximum  in  the  first  case  when  it  is  a 
minimum  in  the  second.  If,  therefore,  e.m.fs.  can  be  induced 
by  the  swinging  coil,  in  the  former  case  it  may  be  expected  that 
a  lagging  current  will  result,  while  in  the  latter  case  a  leading 
current  will  be  induced. 

In  one-half  of  a  period,  the  low-frequency  exciting  current 
rises  from  zero  to  a  maximum  and  declines  to  zero;  during  this 
same  interval  of  time  the  magnet  NS  swings  from  its  position  of 
equilibrium  in  the  plane  YY  to  its  extreme  right  position  and 
back  again  to  its  position  of  equilibrium.  (Fig.  161.) 

The  maximum  velocity  with  which  the  magnet  sweeps  by 
the  low-frequency  exciting  winding  is  reached  when  the  magnet 
passes  its  plane  of  equilibrium.  A  maximum  of  kinetic  energy 


THE  POWER  FACTOR  OF  INDUCTION  MOTORS 


191 


is  stored  in  the  magnet  at  this  time  which  is  given  up  to  the  low- 
frequency  exciting  winding  during  the  swing  of  the  magnet  to 
its  extreme  right  position.  Thus,  energy  is  transferred  from 
the  moving  coil  to  the  exciting  circuit,  the  induced  e.m.f.  de- 
creasing from  a  maximum  to  zero  while  the  exciting  current 
increases  from  zero  to  a  maximum. 

On  the  return  swing  the  low-frequency  exciting  circuit  transfers 
energy  to  the  magnet  which  had  yielded  up  all  its  energy  when 
it  reached  its  extreme  right  position.  Thus,  the  magnet  induces 


Excitatio 


FIG.   161. — Principle   of   Leblanc    "Recuperator"    and    Kapp    "Vibrator." 

an  e.m.f.  in  the  exciting  winding  which  is  in  leading  quadrature 
with  the  exciting  current. 

If  the  magnet  were  allowed  to  swing  beyond  the  x-axis,  it 
would  operate  eventually  as  a  synchronous  motor. 

In  fact,  the  " recuperator"  is  identical  with  an  over-excited 
synchronous  motor.  If  the  mass  of  the  magnet  was  great  and 
the  field  weak,  it  would  act  like  an  under-excited  synchronous 
motor.  Rotation  is  merely  a  special  case  of  the  phenomenon  of 
oscillation. 

The  elementary  theory,  omitting  the  effect  of  mechanical  or 
electrical  damping,  in  the  oscillations  of  the  magnet,  may  be 
given  as  follows : 


192  INDUCTION  MOTOR 

Let  the  moment  of  inertia  of  the  magnet  in  cm  units  be  7.  Let 
6  be  the  angle  which  at  any  moment  of  time  the  magnetic  axis  of 
the  oscillating  body  forms  with  the  magnetic  axis  of  the  stator. 
The  stator  is  excited  with  low-frequency  currents  obtained  from 
the  slip  rings  of  the  induction  motor  in  the  circuit  of  which  the 
"recuperator"  is  connected. 

Let  -JT  be  the  angular  velocity  of  the  swinging  magnet.     This 

magnet  swings  in  such  a  manner  that  it  moves  through  a  total 
angle  of  40mo*  in  the  time  T,  in  which  a  complete  cycle  is  passed 
through  by  the  low-frequency  exciting  currents. 

The  mean  angular  velocity  of  the  rotor  is  therefore  immediately 
obvious  since 

/rlf)\ 

T  —  4-0 

f.firt.     •*-       ~  ~     ~t/  rnn.r. 


Hence, 

46max 
-co 


dt/  m  27T 

Also,  as  the  oscillations  are  assumed  to  take  place  according 
to  a  simple  sine  law, 


<dt, 

D'Alembert's  principle  applied  to  a  rigid  body  rotating  about 
an  axis  is 

7  -^    =   Pmax'Sm  CoZ-d 

where  -575  is  the  angular  acceleration,  Pma*  the  maximum  value 

of  the  force  exerted  by  the  low  frequency  winding  upon  the  swing- 
ing magnet,  and  d  the  arm  of  the  couple  producing  the  angular 
acceleration. 

0  varies  according  to  a  simple  sine  law,  hence 

0    =    0 mas' SHI  Co£ 

de  _ 

dt  ~~    mox'co'cos  w 


THE  POWER  FACTOR  OF  INDUCTION  MOTORS          193 

Therefore, 

I  -jj2  ~  ~  I-0max^2-sm  ul  =Pmax-sin  ut-d 

•      a  *max'd  ftf7f\\ 

-     •  "max    =    ~          j     2  (170) 

Pmax  can  be  calculated  from  the  known  mechanical  force 
acting  upon  a  conductor  in  a  magnetic  field.  Let  B  be  the  den- 
sity of  the  flux  in  the  air-gap,  i  the  effective  low-frequency  excit- 
ing current,  I  the  length  of  the  conductor  in  the  field,  and  z  the 
total  number  of  conductors  in  the  low-frequency  exciting  wind- 
ing exposed  to  the  flux  of  the  magnet  in  the  air-gap. 

Then 


D  D          7 

Pmax    =    g  5l(j~  981 

Therefore,  by  substitution  of  this  value  into  (170)  we  obtain 


2  10-981-/-W2 

The  maximum  value  of  the  e.m.f.  induced  by  the  oscillating 
magnet  in  the  low-frequency  exciting  winding  is 


^ u  ~u~& '  J. \j  vOluS 


/dd\ 
\dt)  max  = 


Bmax'te 

\  an  - 
Substituting  we  obtain 

d 

E  =  6max-u-~-B'l'Z'lQ~*  volts 
z 


TWRRT1 10-8  volts 

And  the  effective  e.m.f. 

-10~8  volts 


This  equation  shows  that  the  magnitude  of  the  injected  e.m.f. 
increases  directly  as  the  product  of  the  square  of  the  flux  to  which 
the  low-frequency  winding  is  exposed  into  the  value  of  the  low- 
frequency  exciting  current,  and  inversely  as  the  product  of  the 
moment  of  inertia  of  the  magnet  into  the  low  frequency. 

Write  for  (171) 

e  = ^ -.K  volts 

l(~i  —  ~*) 

and  we  see  that  the  secondary  phase  angle  varies,  as  the  e.m.f. 

13 


194 


INDUCTION  MOTOR 


induced  in  the  rotor  to  overcome  the  resistance  drop  is  propor- 
tional to  the  slip  frequency  ^i  —  ^2- 

Let  C  be  the  capacity  in  farads  of  a  condenser,  then 

i  =  u-C-e  farads  (171) 


Source  of  B.C. 


FIG.   162. — Connections  of  Kapp  "Vibrator." 

Comparing  this  formula  with  equation  (171)  we  obtain  for  the 
capacity  effect  in  farads  of  this  "recuperator"  the  value 
4-9810  I 


C  = 


10~8 farads 


FIG.  163. — Kapp    "Vibrator"    for    1,500    h.p.  three-phase    induction     motor. 
(Built  by  Westinghouse  Company  of  Pittsburgh.) 

and  the  " elastance,"  or  the  ability  to  let  current  pass 
i  ==  4-9810-7  10~8 


THE  POWER  FACTOR  OF  INDUCTION  MOTORS 


195 


It  is  interesting  to  note  that  the  effect  of  this  mechanical 
condenser  is  such  that  the  secondary  phase  angle  decreases  with 
increasing  slip.  But,  as  it  also  depends  upon  the  secondary 
current  for  its  action,  it  is  not  very  efficient  at  low  slips.  At 
synchronism  it  acts  as  an  impedance. 


FIG.   164. — Polar  diagram  of  1,500  h.p.  induction  motor,  2,200  volts,  three-phase, 
60  cycles,  20  poles.     With  and  without  Kapp  vibrator. 


D.  THE  SAME  METHOD  AS  ELABORATED  BY  G.  KAPP 

Mr.  Kapp  re-invented  the  Leblanc  scheme  and  he  applied  it  in 
practical  form.  He  called  the  device  a  " vibrator."  The 
general  theory  is  identical  with  that  of  the  " recuperator"  of 
Leblanc's. 

Figure  163  shows  a  photograph  of  one  of  these  three-armature 
"vibrators"  built  by  the  Westinghouse  Company  in  connec- 


196 


INDUCTION  MOTOR 


tion  with  a  1,500  hp.  induction  motor.  The  general  principle  of 
the  " vibrator"  is  given  in  Figs.  161  and  162,  which  are  taken 
from  Mr.  Kapp's  paper.1  The  magnet  consists  of  an  armature 
of  small  diameter  wound  like  a  direct-current  armature.  Brushes 
convey  current  to  a  commutator. 

The  polar  diagram  of  this  motor  with  and  without  the  "  vi- 
brator" is  given  in  Fig.  164.  The  field  of  application  of  these 
ingenious  devices  appears  limited. 

BIBLIOGRAPHY 

HORACE  LAMB:  "Dynamics."     Cambridge  University  Press,  1914,  p.  282. 
Par.  95,  "Forced  Oscillations." 


R 


M     Main  Motor 

E     Exciter  with  Laminated  Field 

R     Rheostat 

Fio.  165. — The  Danielson-Burke  method  of  changing  an  induction  motor  into  a 

synchronous  motor. 

HORACE  LAMB:  "The  Dynamical  Theory  of  Sound."  London,  EDWARD 
ARNOLD,  1910,  p.  16.  Par.  8,  et  seq. 

JOHN  PERRY:  "The  Calculus  for  Engineers."  London,  EDWARD  ARNOLD, 
1899,  p.  213.  Par.  148,  et  seq. 

J.  PRESCOTT:  "Mechanics  of  Particles  and  Rigid  Bodies."  London,  Long- 
mans, Green  &  Company,  1913,  p.  328.  Par.  340. 

EDWARD  JOHN  ROUTH:  "Dynamics  of  a  System  of  Rigid  Bodies."  London, 
Macmillan  &  Co.,  Ltd.,  1897,  p.  344.  Par.  432. 

1  G.  KAPP,  The  Electrician,  May  17  and  24,  1912.  "On  Phase  Advancers 
for  Non-synchronous  Machines."  Also  U.  S.  Patents  No.  1,236,716,  Aug. 
14,  1917  and  No.  1,258,577,  March  5,  1918. 


THE  POWER  FACTOR  OF  INDUCTION  MOTORS          197 

HENRI  BOUASSE:  "Coursde  Mecanique  Rationelle."    Paris,  C.  DELAGRAVE, 

Chap.  VII. 
A.  GUILLET:     "Proprietes  Cinematiques  Fondamentales  des  Vibrations." 

Paris,  Gauthier-Villars,    1913. 

E.  THE  DANIELSON-BURKE  METHOD  OF  CHANGING  THE  INDUC- 
TION MOTOR  INTO  A  SYNCHRONOUS  MOTOR 

A  modification  of  the  Danielson  idea,  already  mentioned  by 
Tesla,  of  exciting  a  non-synchronous  motor  with  direct  current 
when  it  approaches  synchronous  speed,  is  due  to  Mr.  James 
Burke. 

An  exciter  with  laminated  field  is  connected  as  indicated  in 
Fig.  165.  When  the  starting  resistance  is  cut  out,  the  exciter 
automatically  functions  as  a  direct-current  generator. 


CHAPTER  XVI 
THE  MAGNETIC  PULL  WITH  DISPLACED  ROTOR 

Before  closing  the  subject  of  poly-phase  induction  motors,  we 
shall  briefly  refer  to  an  important  mechanical  relation.  The 
small  air-gaps  necessary  with  these  motors  make  it  important  to 
be  able  to  forecast  the  amount  of  unbalanced  magnetic  pull  due 
to  excentric  position  of  the  rotor  within  its  stator. 

A.  THE  FORMULA  OF  B.  A.  BEHREND 

The  author  developed  a  very  simple  mathematical  relation, 
giving  the  magnetic  pull  for  a  displaced  rotor,  in  a  paper1  pre- 
sented before  the  American  Institute  in  November,  1900. 


FIG.  166. — The  magnetic  pull  with  displaced  rotor. 

Consider  two  cylinders  placed  eccentrically  as  shown  in  Fig. 
166.  All  around  the  circumference  acting  along  a  radius  vector, 
we  postulate  the  same  m.m.f.  and  assume  the  lines  of  induction 

1  B.  A.  BEHREND,  "On  the  Mechanical  Forces  in  Dynamos  Caused  by 
Magnetic  Attraction."  Trans.,  A.  I.  E.  E.,  Nov.  23,  1900,  p.  617. 

198 


THE  MAGNETIC  PULL  WITH  DISPLACED  ROTOR       199 

to  follow  the  shortest  path.     Their  density  is  then  inversely 
proportional  to  the  length  of  the  gap. 
From  Fig.  166 

da  =  8  —  A-  cos  a 


A   = 


2 
Hence, 

B^        da  _  d  —  A  cos  a 
lBa  ~  ~5    :  6 

Ba  =  B-  -±-  (172) 

1  —  —  •  COS  a 

0 

The  magnetic  attraction  acting  along  a  radius  vector  upon  a 
surface  element  of  the  area  bRda,  in  which  b  is  the  width  of  the 
laminations,  is 

dfc  =  \-Bl-b-R-da 

O7T 

| 

As  the  horizontal  components  of  this  force  on  either  side  of  the 
line  of  symmetry  balance  each  other,  the  remaining  vertical 
component  is 

df  =  ^-Ba-b'R-da-cos  a 

O7T 

Substituting  for  Ba  its  value  above 

=   1 

O7T 


/  '  A  \ 

(1-^-cosa) 
=  co 


-r  cos  a  ) 

0  I 


[A  ~|  2 

—  cos  a.\    as  a  small  quantity,  we  obtain 


STT        '  2A 

1 cos  a 

5 

,    2A 

,  1  +  -r-  cos  a 

0 


=  J-  6BB2  [l  +  ^cosalcos  ada          (175) 
oir  o 


200 


INDUCTION  MOTOR 


By  integrating  this  expression  between  the  limits  0  and  ~  we 

find  the  vertical  component  of  the  magnetic  attraction  in  one 
quadrant. 


T  1C 

2  2 

I  df  =  ^bRB*  I   (l  +  ^ycos  «)  cos  ada 
Jo  Jo 


cos  ada  H — -  I  cos2  ada 
o    E 


(176) 


(177) 


sn  a 


.  2A  1    .     0 

-f—  •-;  sm  2a 


6   4 


IT  7T 

2  2 

2Aa 

"*"   5  '2 


2A    TT-I 
5     4J 


(178) 


For  the  vertical  component  directed  downwards,  we  find  simi- 
larly the  expression 


Hence, 


(180) 


The  total  force  is  twice  as  great,  as  we  have  integrated  only 
over  a  quadrant,  therefore 

•«  r>  A 

(181) 
(182) 


Z  =       B*A      dynes 


irRb 
db-da 


Numerical  Example: 


B  =  5,900  c.g.s. 
A  =  9,350  cm.2 
A  =  0.1  cm. 
d  =  0.36  cm. 

5,9002X  9,350  2X0.1 


Z  = 


8 


0.36 


dynes 


9.81  X  105  dynes  =  1  kg. 

=  7,400  kg. 


THE  MAGNETIC  PULL  WITH  DISPLACED  ROTOR       201 
B.  THE  ACCURATE  SOLUTION  BY  J.  K.  SUMEC 

J.  K.  Sumec1  has  given  the  integral  of  equation  (174)  and 
obtained  an  elegant  solution. 

He  starts  with  the  author's  equation  (174). 


The  vertical  component  is 

2  - 

'  ' 


Taking  points  diametrically  opposite  and  simplifying 

4-  cos2  a 
cos  a  cos  a  5 

2 


(A  \2/         A  \'2r          /A\  2  n 

!--«*«)  (1+-C080,}  [l-(-)cOS*«J 


A  cos2 


i -$•-•] 


A  A  „    /  cos'  ^ 


Substituting 

u  =  tga 
du  =  (1  -+ 

1  1 


cos2a 


I 


1  +  tgza        1  +  u2 
COS2adct 


(185) 


o 
This  integral  is  a  rational  algebraic  function  of  the  form 


/» 

«/  o 


a  =  J  ~  (t)  2  (186) 


o  . 
1  J.  K.  SUMEC,  Zeitschrift  fur  Elektrolechnik,  Vienna,  Dec.  18,  1904. 


202 


INDUCTION  MOTOR 


which  can  be  solved  readily  by  trigonometric  substitution 

x  =  Va  tan  0  (187) 

dx  =  V~a  ^L  (188) 


Va-  d0  I 

(a  +  a^Y^e=     I 

V  COS207  JQ 


COS20 

\/a-  cos20-d0 


(189) 


=  cos20-d0 
a 


|-  sin  20] 


I 


£-\-^=  tan-1  -4=  +  — r-il  I  (190) 

2alV«  Va       a  +  x2ir 


1      TT 


Substituting  this  evaluated  integral  in  (184) 

52.  2A  1 


(191) 


(192) 


(193) 


This  is  Prof.  J.  K.  Sumec's  very  elegant  result  which  but  for 
the  last  term  is  identical  with  the  author's  formula  (181). 
The  factor 


['  -  e) : 


is  evaluated  for  different  ratios  of  -r  and  the  following  table  is 
obtained  which  is  also  taken  from  Prof.  Sumec's  paper. 


A 
8 

0.1 

0.2 

0.3 

0.4 

0.5 

1 

1.015 

1.063 

1.152 

1.30 

1.54 

[•-©']" 

THE  MAGNETIC  PULL  WITH  DISPLACED  ROTOR       203 

For  eccentricities  less  than  25  per  cent  the  errors  in  using  the 
author's  formula  are  negligible.  At  30  per  cent  eccentricity 
the  error  is  15  per  cent,  and  at  50  per  cent  eccentricity,  which  is 
not  at  all  unusual  in  induction  motors,  the  error  is  54  per  cent. 
On  this  account  this  subject  has  been  included  in  this  book. 

BIBLIOGRAPHY 

The  paper  by  E.  ROSENBERG,  Trans.  A.  I.  E.  E.,  1918,  Vol.  XXXVII, 
p.  1417,  with  a  full  bibliography  by  ALEXANDER  MILLER  GRAY  and  J.  G. 
PERTSCH,  JR.,  entitled  "Critical  Review  of  the  Bibliography  on  Unbalanced 
Magnetic  Pull  in  Dynamo-electric  Machines,"  should  be  consulted  by  the 
reader.  Also  the  papers  by  B.  A.  BEHREND,  "On  the  Mechanical  Forces  in 
Dynamos  Caused  by  Magnetic  Attraction,"  Trans.  A.  I.  E.  E.}  Nov.  23, 
1900,  p.  617.  And  J.  K.  SUMEC,  loc.  tit. 


CHAPTER  XVII 
THE  SINGLE-PHASE  INDUCTION  MOTOR 

"The  single-phase  motor  has  been  the  subject  of  perhaps  more  theo- 
retical speculation  than  any  other  dynamo-electric  machine,  and  the 
reason  for  this  is,  undoubtedly  that  it  is  in  its  functioning,  the  most 
complicated  of  all  dynamo-electric  machines,  although  in  its  structure  it 
is  the  simplest  of  them  all."1 

The  simplified  and  improved  transformer  diagram  which  has 
served  us  well  in  the  understanding  of  the  phenomena  in  poly- 
phase motors,  serves  our  purpose  equally  well  in  the  treatment 
of  the  single-phase  motor.  We  shall  employ  two  methods  of 
dealing  with  this  problem. 

A.  Galileo  Ferraris  and  Andre  Blondel  have  made  use  of 
FresneFs  theorem  that  an  alternating  or  oscillating  field  of  force 
may  be  replaced  by  two  equal  and  oppositely  rotating  fields  the 
amplitude  of  each  of  which  being  equal  to  one-half  the  maximum 
amplitude  of  the  alternating  field.  These  two  fields  rotate  in 
opposite  directions  at  an  angular  velocity  equal  to  2?r  times  the 
frequency  of  the  alternating  field,  assuming  a  two-pole  field  as 
has  been  done  throughout  this  volume. 

Employing  Fresnel's  theorem,  the  author2  developed  the 
following  simple  vector  diagram  which  has  proved  amply  accurate 
for  all  the  purposes  of  engineering  applications. 

A  two-pole  rotor,  revolving  at  an  angular  velocity  correspond- 
ing to  a  frequency  ^2  in  an  oscillating  magnetic  field  whose 
frequency  of  oscillation  is  ^i,  has  relative  to  field  I  of  Fresnel's 

two  fields  a  slip  —          — -  =  s  and  relative  to  the  other  field  II  a 

~i 

/*>"/i    — |—   ^*^9 

slip  — --    —  =  2-  s. 

~i 

Consider  the  field  II.     At  the  great  slip  —  -  the  primary 

^i 

and  secondary  ampere-turns  act  almost  in  space  opposition. 
They  would  act  exactly  in  space  opposition  if  the  ohmic  rotor 

1  E.  F.  W.  ALEXANDERSON,  Trans.  A.  I.  E.  E.,  Part  I,  p.  691,  1918. 

2  B.  A.  BEHREND,  "Asynchronous  Alternating  Current  Motors,"  E.  T.  Z., 
March  25,  1897. 

204 


.THE  SINGLE-PHASE  INDUCTION  MOTOR  205 

resistance  could  be  neglected.  In  order  to  simplify  the  under- 
standing of  the  theory  of  this  motor  we  shall  assume,. which  is 
admissible  without  great  error,  that  so  far  as  the  field  II  is 
concerned  the  rotor  resistance  is  negligible  and  rotor  and  stator 
ampere-turns  are  in  phase  in  space,  the  stator  ampere-turns 
being  just  enough  larger  than  the  rotor  ampere-turns  to  magnetize 
the  core  to  the  extent  of  producing  a  field  which  balances  the 
voltage  required  to  pass  through  the  stator  of  field  II  the  current 
the  ampere-turns  of  which  we  have  considered  here. 

The  primary  ampere-turns  of  the  single-phase  motor  have 
been  resolved  into  two  oppositely  rotating  components  of  one- 
half  the  amplitude.  This  resolution  is  equivalent  to  two  poly-phase 
motors  whose  stator  windings  are  connected  in  series  while  the 
rotor  windings  are  common  to  both  stators.  The  rotor  being  in- 
tegral, the  rotor  torques  act  in  opposite  directions. 

The  field  I  tries  to  turn  the  rotor  in  a  clockwise  direction, 
while  the  field  II  tries  to  turn  the  rotor  in  a  counter-clockwise 
direction.  The  motor  I  takes  the  larger  share  of  the  voltage 
since  the  apparent  reactance  of  an  induction  motor  is  high  at  a 
small  slip,  and  low  for  a  large  slip.  Hence,  as  the  stators  of  I 
and  II  carry  the  same  current,  viz.,  one-half  the  total  current, 
the  voltage  impressed  on  the  stator  of  motor  II  is  very  small. 

If  we  neglect  in  the  operation  of  motor  II  the  rotor  resistance, 
it  has  been  pointed  out  that  the  effect  of  motor  II  consists  mainly 
in  the  reactance  effect  of  this  motor.  The  voltage  impressed  upon 
motor  I  is  diminished  by  the  voltage  required  by  motor  II  and 
this  voltage  is  always  proportional  to  the  stator  current  of  motor 
I  and  in  time  quadrature  with  it. 

Therefore,  the  effect  of  motor  II  may  be  taken  into  account 
by  correspondingly  increasing  the  primary  leakage  of  motor  I. 
The  resultant  vector  diagram  of  a  single-phase  induction  motor 
may  now  be  developed. 

a.  THE  MAGNETIZING  AND  NO-LOAD  CURRENTS 

/"X^/1        -H—       /^M^O 

At  synchronism,  the  slip  of  motor  II  is — =  2.     The 

same  current  passes  through  the  stators  of  motor  I  and  II.  With 
no  leakage  and  no  resistance  in  the  rotor  and  stator  of  motor  II, 
the  voltage  impressed  upon  motor  II  would  be  zero  and  therefore, 
the  full  voltage  would  be  impressed  upon  motor  I.  Hence,  the 
magnetizing  current  would  be  exactly  doubled. 


206  INDUCTION  MOTOR 

As  the  magnetizing  current  is  proportional  to  the  impressed 
voltage  it  is  admissible  to  write  for  the  case  above,  designating 
the  magnetizing  current  of  the  single-phase  induction  motor  by 
z'M  and  its  no-load  current  by  i0 

i0  =  2iM  +  O  =  2tM  (195) 

where  z'M  is  the  magnetizing  current  of  either  motor  I  or  II. 

However,  taking  leakage  into  account,  we  have  the  current 

2  passing  through  each  stator  I  and  II.     The  impressed  voltage 


•fx 


(?,-!)  tsCV 


(«,-*>*/< 


Fio.  167.  —  Vector  diagram  of  the  two-motor  theory  of  the  single-phase  induction 
motor.     (The  two  motors  in  series.) 

on  motor  I  at  no  load  is  proportional  to  „",  while  the  impressed 

voltage  on  motor  II  is  proportional  to  2",  where  z'J,1  is  the  magnet- 
izing current  of  motor  II.  Both  magnetizing  currents  must 
equal  the  magnetizing  current  of  the  single-phase  induction 
motor. 

We  shall  draw  a  flux  diagram  and  a  current  diagram  to  make 
this  clear.  These  diagrams  may  be  compared  advantageously 
with  the  corresponding  diagrams  for  the  equivalent  circuits. 

An  examination  of  the  flux  and  current  diagrams  reveals 
the  following  simple  relations:  (Figs  167,  168,  169,  170,  171, 
and  172). 


THE  SINGLE-PHASE  INDUCTION  MOTOR  207 


OK  = 


BG  = 


FIG.  168. — The  time-phase  vector  diagram  of  the  fluxes  of  the  single  phase 
induction  motor.     The  two  motor  theory. 


FIG.  169. — The    circle    diagram    of    the    single-phase    induction    motor.     The 
two-motor  theory  (after  Behrend). 


208 


INDUCTION  MOTOR 


l/2-X2 


FIG.  170. — The    complete   and    theoretically   exact   equivalent   circuits   of   the 
single-phase  induction  motor  in  the  two  motor  theory. 


FIG.   171. — First  approximation  of  equivalent  circuits  of  single-phase  induction 
motor  in  the  two  motor  theory. 


THE  SINGLE-PHASE  INDUCTION  MOTOR 


209 


The  ratio  of  the  magnetizing  current  of  the  second  motor  i" 
to  the  total  current  of  this  motor  at  no  load  is  i"  -f-  5-.     Now,  tf : 


2   is  equal  to  the  ratio  of  the  sum  of  the  leakage  fields  of  the 
second  motor  to  the  total  flux  %$\v\.     Hence, 


f\l  =  ±(Vl  - 


•       (196) 

=  0(^2  —  1)—  approximate 
£  Vi 

(197) 


FIG.  172. — Second  approximation  of  equivalent  circuits  of  single-phase 
induction  motor  in  the  two  motor  theory.  Admittance  y  negligible  in  comparison 
with  1/0:2. 


Therefore, 


-  1 


(198) 


assuming  vi  =  vz  which  is  permissible  for  the  second  motor. 
A  simple  transformation  now  yields 


.„ 


(199) 


The  total  magnetizing  current  of  the  single-phase  motor  is  the 

14 


210  INDUCTION  MOTOR 

sum  of  the  magnetizing  currents  of  motor  I  and  motor  II. 

*.  -       -  >"  '    (2°°) 


In  speaking  loosely  of  "adding  magnetizing  currents"  of  two 
motors  in  series,  we  mean  of  course  that  the  voltages  impressed 
upon  each  motor  may  be  added  and  to  the  sum  of  these  voltages 
corresponds  a  magnetizing  current  equal  to  the  sum  of  the 
magnetizing  currents  of  each  motor. 

The  same  result  is  readily  read  off  the  Fig.  169  if  we  divide  OD 
by  OC.     We  obtain 

OD  +  OC  =  (zvi  -  -}-  =  (2»i!;a  -  1) 


For  (7  =  0,  that  is,  for  a  motor  without  leakage,  we  have: 
A  common  value  for  a  is  0.05  when 
1 

£/i  -*-  IQ  =  2 

The  point  G,  Figs.  168  and  169,  is  determined  in  the  same 
manner  as  in  the  theory  of  the  general  alternating  current 
transformer,  from  the  consideration  that  the  ratio  between  the 
magnetizing  current,  proportional  to  OB,  and  BG,  is  equal  to  a- 

=   ViVz   —    1. 

It  is  clear  that  the  point  C  moves  also  on  a  circle  as  the  point 
D,  which  moves  on  the  circumference  of  the  circle  BHD,  divides 
OC  in  a  constant  ratio.  (Fig.  168.) 

Thus  the  locus  of  C  is  the  circle  KGC,  where 

(203) 


and  OB  =  i»  (204) 

and  BG  =  ip  +  <r  (205) 

We  have  re-drawn  the  current  diagram  for  these  conditions 
in  order  to  impress  the  picture  more  vividly  upon  the  mind,  Fig. 
169. 


THE  SINGLE-PHASE  INDUCTION  MOTOR  211 

b.  THE  CURRENTS  IN  THE  ARMATURE 

The  advantages  for  popular  use  of  this  somewhat  loosely- 
knit  and  so  much  misunderstood  theory  of  two  poly-phase 
motors  connected  in  series  to  simulate  the  single-phase  induc- 
tion motor,  may  now  be  discussed  in  more  detail.  All  manner  of 
errors  and  mistakes  have  been  made,  even  by  leading  writers, 
in  the  interpretation  of  the  theory  by  means  of  the  two-motor 
method.  Twenty-six  years  ago,  in  a  famous  and  otherwise 
brilliant  book,  an  author  assumed  that  the  two  poly-phase 
motors  were  connected  in  parallel  and  this  same  curious  mistake 
has  recently  been  reproduced  in  a  noted  textbook  and  also  in 
handbook  of  wide  circulation.  But  such  misapprehensions  are 
not  inherent  in  the  theory. 

The  current  in  the  armature  of  the  single-phase  motor  is 
equal  to  the  vector  sum  of  the  secondary  currents  Vi-BD  and  v\* 
DC  of  the  two  poly-phase  motors,  hence,  it  is  equal  to  v\  BC. 

A  glance  at  Fig.  169  shows  that  only  about  one-half  of  the 
energy  dissipated  in  the  armature  can  be  utilized  for  the  produc- 
tion of  the  torque.  At  all  loads  the  secondary  currents  repre- 
sented by  BD  and  DC  remain  very  nearly  equal  and,  as  only  one 
motor  is  doing  useful  work,  the  armature  currents  in  motor  II 
represent  a  loss  very  nearly  equal  to  the  loss  in  the  armature  of 
the  working  motor  I.  The  slip  in  the  single-phase  motor  indi- 
cates, therefore,  only  about  one-half  of  the  energy  dissipated  in  the  ar- 
mature, hence,  the  loss  in  the  rotor  of  a  single-phase  motor  is  twice 
as  large  as  that  in  a  poly-phase  motor,  provided  the  slip  be  equal 
in  the  two  motors. 

It  is  also  interesting  to  note  that,  at  synchronism,  the  rotor 
carries  a  current  of  double  frequency.  At  speeds  lower  than 
synchronism  a  fundamental  of  slip  frequency  is  superimposed 

^^  1    ~\~   ^^^2   ^ 

upon  the  frequency  of  - 

^i 

c.  THE  TORQUE  AND  SLIP 

The  torque  can  be  calculated  as  follows:  Suppose  the 
armature  is  wound  in  three  phases,  each  having  the  resistance 
r2.  The  output  of  the  motor  is  then: 

P  =  eiii-coafri  -  ii2n  -  3*22r2  (206) 

1  M.  I.  PUPIN  is  right,  Trans.  A.  I.  E.  E.,  1918,  p.  686,  that  it  is  not 
necessary  to  "assume  two  rotary  fields  produced  by  the  stator  current  at 
all;  in  fact  they  have  no  physical  existence;  but  the  presence  of  two  rotary 
magnetic  fields  produced  by  the  rotor  current  is  a  fact." 


212  INDUCTION  MOTOR 

from  which  follows 

fil  fiD    ,      —  .P  f9O7^ 

Ul.Uiytfifcg    —  -Lwatta  IZU/  J 

~2 

in  which  equation  Dmkg  is  the  torque  in  mkg  and  p  the  number 
of  north  or  south  poles. 

The  following  reasoning  yields  a  value  for  ^2: 

We  have  for  motor  I, 

9.81  ZMcoi  -  o>2)  =  ~^  (208) 

and  for  motor  II 

9.81-Z>n(ui  +  co2)  =  ^~  (209) 

where  coi  =  2 — -  and 
P 

co2  =  27T-n  the  angular  velocity  of  the  rotor  at  n  revolutions 
per  second. 

Hence, 

9.81  (Di  -  Z>n)«2  =  3^2V2-         \  (210) 

(211) 

L  x_2/  (212) 

Writing  S  for  —  we  obtain 

r  1  n 

(213) 


To  illustrate,  Let  us  assume  ~i  =  50,  and  ~2  =  45,  then  we 
have 


=  (j.Zo  r  watts 

In  words,  if  the  slip  is  10  per  cent,  the  loss  of  energy  in  the 
armature  amounts  to  23  per  cent  of  P  or  approximately  twice 
the  percentage  of  the  slip,  if  figured  upon  P  +  3i'2V2. 

It  is  instructive  to  compare  (212)  with  the  similar  one  in  the 
poly-phase  induction  motor,  which  develops  after  some  trans- 
formations : 

"  " 

(214) 


THE  SINGLE-PHASE  INDUCTION  MOTOR 


213 


In  words  the  slip  in  per  cent  is  equal  to  the  ratio  of  the  armature 

/"*~/ 1 
loss  to  the  entire  secondary  power P. 

~2 

In  Fig.  173  the  output  and  torque  in  watts  and  synchronous 
watts  as  a  function  of  the  rotor  speed  expressed  in  cycles  per 
second  have  been  represented  as  calculated  from  the  results 
given  in  this  chapter.  The  heavy  lines  represent  a  rotor  with 
small  resistance,  while  the  broken  lines  represent  a  rotor  with 
fairly  large  resistance.  These  curves  represent  actual  conditions. 


250 
200 
150 


FIG.  173. — Output  and   torque   curves   of   single-phase   induction   motor   as   a 
function  of  the  speed. 


d.  EXPERIMENTAL  DATA 

There  are  reproduced  here,  from  the  first  edition  of  this 
book,  the  characteristics  of  a  10-hp.  single-phase  motor  for 
110  volts,  50~,  and  1,500  r.p.m.  The  total  number  of 
conductors  in  the  field  was  120;  the  total  number  of  conductors 
in  the  armature  was  312.  The  resistance  of  the  field  was  0.015 
ohm;  of  each  of  the  three  phases  of  the  armature,  it  was  0.08 
ohm. 

It  is  instructive  to  calculate  the  armature  loss  in  watts  for 
the  greatest  load.  We  find  this  to  be  552.024  =  730  watts, 
corresponding,  according  to  equation  (213),  to  a  slip  of  2.1  per 
cent.  The  discrepancy  between  this  and  the  measured  value  of 
2.66  per  cent,  is  due  probably  mainly  to  the  difficulty  of  measur- 
ing a  small  slip. 


214 


INDUCTION  MOTOR 
TESTS  OF  10-HP.  MOTOR 


o 

£ 

b 

C 

u 

d} 

CQ      ,-* 

,^ 

2 

£ 

£ 
0 
cr 

B> 

a2 

11 

02  Is 

•^     D- 

£     -*» 

1 

o 

si 

ft 

2 

.2*  o> 

S    frt 

g  £ 

^     ff 

03      & 

£ 

o   °3 

tf 

GQ 

•< 

<!  ^ 

^  pSI 

^      C 

i 

dn  " 

0 

51.5 

100.00 

29.0 

0 

900 

0 

0.000 

0.283 

1,553 

51.5 

51.5 

1,700 

o 

0.000 

0  .  300 

1,579 

52.8 

54.0 

2,250 

700 

0.310 

0.379 

1,572 

52.4 



58.0 

.  . 

3,300 

1,770 

0.536 

0.518 

,554 

52.4 

90.0 

8,100 

6,300 

0.778 

0.820 

,547 

51.4 

112.0 

21 

10,500 

8,100 

0.770 

0.850 

,525 

51.5 

1.30 

138.0 

29 

13,500 

10,600 

0.785 

0.890 

,517 

50.8 

0.52 

150.0 

31 

14,700 

11,400 

0.775 

0.890 

,483 

50.2 

1.80 

169.0 

35 

16,500 

13,000 

0.788 

0.888 

1,430 

48.8 

2.40 

200.0 

.  . 

19,400 

14,860 

0.765 

0.882 

1,460 

49.9 

2.66 

245.0 

55 

23,500 

16,830 

0.716 

0.873 

e.    CALCULATION     OF    THE    MAGNETIZING    CURRENT    OF    THE 
SINGLE-PHASE  MOTOR 

We  must  now  calculate  the  counter  e.m.f .  of  the  single-phase 
motor  induced  upon  itself  by  the  oscillating  resultant  field  in  the 
stator.  We  shall  consider  two  cases,  viz.,  first,  the  case  in  which 


\ 


F-f  5 

fc-l. 


W     \ 


FIG.  174. — The  field  belt  of  the  single-phase  motor. 

the  field  coil  is  spread  over  two-thirds  of  the  pole-pitch;  secondly, 
the  case  in  which  the  field  coil  covers  the  entire  pole-pitch. 

First. — We  saw  in  Chap.  I  that  the  e.m.f.  induced  by  a  field 
F  in  a  coil  of  a  certain  width  is  two-thirds  as  large  as  the  e.m.f. 
which  would  be  induced  by  the  same  field  in  a  coil  which  is  not 
distributed  but  lodged  in  one  slot.  For  the  latter  case,  we  have 

e  =  2.22~-zF-lQ-*  volts 
Figure  174  shows  the  form  of  the  field  for  a  coil  the  width  of  which 


THE  SINGLE-PHASE  INDUCTION  MOTOR 


215 


is  equal  to  two-thirds  of  the  pole-pitch.    Call  the  number  of  active 
conductors  per  pole  n,  then  we  have  for  the  induction  in  the  air 


B 


1.6A 


(215) 


In  this  formula  i^  is  the  effective  value  of  the  magnetizing 
current,  A  the  air-gap  on  one  side.  If  b  is  the  width  of  the  iron 
of  the  motor  in  cm.,  t  the  pole-pitch,  then  we  have, 

2, 


F  = 


(216) 


T 

B 


FIG.   175. — The  field  belt  of  the  single- phase  motor. 

Instead  of  a  coefficient  of  2.22  we  obtain,  as  will  easily  be  seen, 
a  coefficient  1.85.     Thus,  we  have  the  equation 

e  =  1.85-z-F-lO-8  volts  (217) 

Secondly. — Figure   175   shows  this   case.     The   total   flux   is 


For  the  e.m.f .  we  have 


F  =  \b-t.B 


e  =  ~2.22-~-  Z-F-10-* 


(218) 


e  =  lA8-~-z-F-10-*  volts  (219) 

It  is  evident  from  these  equations  that  it  is  not  advantageous  to 
use  too  wide  a  coil  spread. 


CHAPTER  XVIII 
THE  SINGLE-PHASE  INDUCTION  MOTOR,  CONTINUED 

B.  THE  CROSS  FLUX  THEORY 

In  1894  Alfred  Potier1,  and  in  1895  and  1903  H.  Goerges, 
discussed  the  theory  of  the  single-phase  induction  motor  on  the 
basis  of  a  cross  field,  or  speed  field,  generated  by  the  rotation  of 
the  rotor  conductors  in  the  resultant  primary  field  of  the  main 
circuit.  Dr.  A.  S.  McAllister  also  developed  this  theory  and  he 
has  always  been  with  justice  the  advocate  and  protagonist  of 
this  theory  in  America  versus  the  two-motor  theory.  J.  K. 
Sumec,  basing  his  development  of  the  theory  on  the  Blondel 
flux  diagram  and  following  the  present  author's  method  of  reason- 
ing in  connection  with  the  circle  diagram  of  the  poly-phase 
motor,  derived,  in  a  brilliant  article  in  the  Zeitschrift  fur  Elek- 
trotechnik,  No.  36,  Vienna,  1903,  the  correct  circle  diagram  of 
the  single-phase  induction  motor.  In  this  chapter  we  shall 
follow  mostly  the  clear  analysis  of  the  phenomena  as  given 
by  Sumec  and  McAllister.  Though  the  theory  is  not  easy  to 
master,  I  believe  its  study  will  repay  amply  as  Dr.  McAllister's 
contention  is  doubtless  correct  that  the  physical  phenomena2 
are  accounted  for  more  logically  in  this  theory  than  in  the 
' '  two-motor  "  theory. 

(a)  A  GENERAL  CONSIDERATION  OF  THE  THEORY 

A  general  consideration  of  this  theory,  based  on  Fig.  176  with 
slight  modifications  is  taken  from  Dr.  McAllister's  book  and  it  is 
a  good  introduction  to  the  more  complex  analysis.  The  vertical 
set  of  poles  may  represent  the  exciting  winding  on  the  stator,  in 

1  Bulletin  de  la  Societe  Internationale  des  Electritiens,  Paris,  May  1894. 

2  See  also,  J.  SLEPIAN,  Trans.  A.  I.  E.  E.,  1918,  p.  661.     Referring  to  the 
two-motor  theory  Dr.  SLEPT  AN  says:  "  Of  course,  the  instantaneous  torque, 
heating,  etc.,  cannot  be  obtained  in  this  way.     This  points  to  one  difference 
between  the  mechanically  connected  series  machines  and  the  single-phase 
current  motor.     In  the  former,  the  torque  in  each  machine  is  constant  in 
time,  so  that  the  same  is  true  of  their  sum.     In  the  latter  the  torque  is 
pulsating,    vanishing   generally  four  times  per   cycle.     The  mean  value, 
however,  is  the  same  in  the  two  cases." 

216 


THE  SINGLE-PHASE  INDUCTION  MOTOR 


217 


reality  of  course  it  is  a  distributed  drum  winding.  It  sets  up  a 
field  which  is  vertical  in  space.  If  we  consider  the  short-cir- 
cuited rotor  conductors,  then  the  combined  effect  of  the  im- 
pressed m.m.f.  and  of  the  induced  m.m.f.  produces  a  resultant 
field  F2  which  we  shall  consider  here  as  in  the  transformer  and 
induction  motor. 

It  is  perfectly  clear  that,  so  long  as  the  rotor  does  not  move, 
there  cannot  be  any  other  resultant  rotor  field  than  the  field 
F2.  However,  as  soon  as  the  rotor  cuts  through  the  resultant 
field  FZ,  an  e.m.f.  of  rotation  will  be  impressed  upon  its  conduc- 
tors. The  time-phase  of  this  impressed  e.m.f.  must  coincide 


Transformer 


Speed  Field  Axis  or 
X-  Axis  for  Fs 


Space  Phases  of  F2  and   Fa 


FIG.  176. — Physical  representation  of  the  cross  flux  theory  of  the  single-phase 
induction  motor.     (After  McAllister). 

with  the  time-phase  of  F2.  Hence,  if  any  current  were  to  flow 
in  the  short-circuited  rotor  conductors,  this  current  would  pro- 
duce a  field  whose  space-phase  was  horizontal.  Therefore,  it  is 
now  clear  that  the  rotation  must  produce  such  a  field  which  is 
called  the  "cross  field"  or  the  " speed  field/'  and  that  the  mag- 
nitude of  this  field,  as  well  as  its  time-phase,  depend  upon  the 
rotor  resistance,  the  rotor  leakage,  and  the  reluctance  of  the 
magnetic  circuit  of  the  speed  field. 

Let  it  be  observed  at  the  outset  that,  whatever  the  time-phase  of 
these  currents,  on  account  of  their  space  quadrature  in  relation  to 
the  main  field  they  no  more  react  upon  this  than  does  one  phase  of  a 
two-phase  motor  react  upon  the  other. 


218 


INDUCTION  MOTOR 


The  induction  produced  by  a  certain  magnetization  is 


B  = 


Px 


where  ix  is  the  current  in  the  "a>system,"  i.e.,  the  speed  field, 
and  px  the  total  reluctance  of  the  speed  field  including  its  leakage 
field.  The  total  flux  follows  directly  from  the  area  and  distri- 
bution of  the  induction. 

If  there  is  no  resistance  in  the  rotor,  or  if  the  reluctance  of  the 
speed  field  circuit  were  zero,  then  the  time-phase  of  the  speed  flux 
Fa  would  be  in  quadrature  with  the  impressed  e.m.f.  of  rotation, 


FIG.  177. — The  e.m.f.'s  in  the  speed  field  and  their  time-phases. 

of  the  speed  field. 


Production 


eFs,  lagging  90  degrees  behind  this  e.m.f.  (We  use  the  subscripts 
FT,  FS,  FSS  and  FST  to  indicate  that  an  e.m.f.  is  due  to  the  flux 
F  or  the  flux  F8,  and  produced  by  transformation  T  or  by  speed 
rotation  s.) 

Exactly  as  in  an  open-circuited  transformer — and  the  speed 
field  is  always  in  the  open-circuited  condition — by  impressing  upon 
the  speed  field  circuit  the  e.m.f.  eFS  a  current  flows  which,  through 
its  rate  of  change,  induces  a  counter  e.m.f.  in  time  quadrature 
and  lagging  90  degrees  behind  the  flux  produced  by  the  current 
which  is  in  time-phase  with  the  flux.  (See  Fig.  177.)  F8  is 


THE  SINGLE-PHASE  INDUCTION  MOTOR  219 

the  speed  field  in  time-phase  with  ix  which  is  the  current.  We 
designate  the  current  with  the  subscript  x  as  its  m.m.f.  acts  in 
space  in  the  X-axis. 

Similarly,  as  in  the  transformer  on  open  circuit, 

OA  =  eFS 
is  the  impressed  e.m.f.  generated  by  rotation  in  the  main  field Fs 

AB  =  ixrz  is  the  ohmic  drop  in  the  rotor. 

AC  =  eFfis  is  the  counter  e.m.f.  produced  by  the  rate  of  change 
of  ix.  This  counter  e.m.f.  may  be  considered  as  composed  of 
two  e.m.f s.  if  we  wish  to  resolve  the  total  field  in  the  speed  field 
or  x-axis  produced  by  ix  into  a  leakage  field  and  an  air-gap  field. 
However,  this  is  entirely  unnecessary  as  the  speed-field  circuit 
is  in  the  position  of  a  choke  coil. 

Let  XQ  be  the  total  reactance  of  the  speed-field  circuit  including 
its  leakage  effects  and  let  it  be  inversely  proportional  to  its 
reluctance  px,  then 

AB  =  €p9T  =  ixXo 
Hence,  OB  +  AB  =  ixrz  -r-  ixx0 

.-.  tgi  =  ??  =  JL.  =  Constant  (220) 

7*2  l°2px 

It  appears,  therefore,  from  this  very  general  inspection,  that 
the  time-phase  angle  £  between  the  speed  e.m.f.  OA  =  eFS  and 
the  speed  flux  Fs  is  constant. 

Since  OA  must  be  in  time-phase  with  F2)  it  appears  that  the 
time-phase  angle  between  the  main  field  and  the  speed  field  is 
constant  at  all  loads  and  speeds. 

Considering  now  the  main  field,  or  the  Y-system  of  the  rotor 
in  space,  we  note  at  once  that  there  will  be  induced  in  it  first, 
an  e.m.f.  of  transformation  eFT  and,  secondly,  an  e.m.f.  of  rotation 
produced  by  the  cutting  of  the  rotor  winding  through  the  speed 
field  F8.  This  e.m.f.  we  denote  by  eFsS.  The  former  is  in  time 
quadrature  with  F2,  while  the  latter  is  in  time-phase  with  F8.  (Fig. 
178.) 

As  eFT  is  due  to  the  rate  of  change  of  F2,  it  lags  in  time  90 
degrees  behind  F2,  while  eFss — produced  by  rotation  in  the  speed 
flux — must  be  opposed  to  eFT,  their  vector  difference  being  equal 
to  the  drop  due  to  the  resistance  and  local  leakage  reactance  of 
the  rotor. 

In  the  theory  of  the  poly-phase  induction  motor  we  have 


220 


INDUCTION  MOTOR 


replaced  with  advantage  the  local  leakage  reactances  by  their 
respective  leakage  fluxes.     The  same  will  be  done  here. 

We  shall  consider  from  now  on,  unless  specifically  stated 
otherwise,  that  F2  represents — as  in  the  theory  of  the  induction 
motor — the  net  flux  actually  passing  through  the  rotor  winding 
so  that,  vectorially,  F2  =  F  —  /2,  where  /2  is  the  local  secondary 
leakage  flux  in  the  transformer  field  or  F-axis. 


—    G  FS  •  sin 


e.m.f.generated  by  speed  rotation 


Fio.   178. — Single-phase   induction  motor.     Time-phase  diagram   of   the   main 
field  and  the  speed  field  and  their  e.m.f's. 

If  this  is  done,  then,  as  in  the  theory  of  the  induction  motor,  we 
may  draw  the  following  diagram  in  which  the  resultant  of  eFS 
and  eFsT  is  used  to  overcome  only  the  resistance  of  the  rotor 
winding  the  same  applying  to  the  resultant  of  eFT  and  ePaS. 

Very  simple  relations  can  now  be  derived  for  the  speed  flux 
in  terms  of  the  main  flux.  These  follow  directly  from  the 
geometrical  figure.  At  standstill  A^L  is  zero  arid  N  lies  at  L.  At 


THE  SINGLE-PHASE  INDUCTION  MOTOR  221 

synchronism  N  lies  at  P  and  above  synchronism  the  motor  turns 
into  a  generator.  (Fig.  178.) 

S  is  the  ratio  of  co2  -r-  coi 

0)1    =    2-7T' — 'i 

The  reader  is  cautioned  that  we  have  designated  the  slip  s  = 

/****'  1          —        /^XN/O  /^XM'O 

with  a  small  s,  and  the  ''speed"  —  with  a  large  $  = 

1  -  s. 

If  the  direction  of  rotation  be  reversed,  i.e.,  if  S  is  made  nega- 
tive, S2  remains  positive  and,  therefore,  the  range  from  L  to  N 
and  to  P  will  be  resumed.  This  is  strikingly  illustrated  by  the  fact 
that  a  single-phase  induction  motor  runs  in  either  direction 
dependent  on  the  direction  of  the  impulse  which  is  necessary 
to  make  it  start. 

(b)  THE  DERIVATION  OF  THE  CIRCLE  DIAGRAM  AND  THE  Locus 
OF  THE  PRIMARY  CURRENT 

It  is  interesting  and  important  to  investigate  the  locus  of  the 
primary  current  under  different  loads  and  speeds.  (Fig.  179) 

Let  ab  be  the  secondary  leakage  flux  in  time  phase  with 

MN  =  iyrz  of  the  previous  figure. 

Let  OA  =  Fz  be  the  NET  transformer  flux,  whose  space- 

phase  is  the     -axis.' 

Let  be  =  NM  =  3>2  be  the  " fictitious"  secondary  flux 

proportional  to  iy  (iy  corresponds  to  iz  in  the  theory 
of  the  transformer  and  induction  motor). 
ab  -r-  be  =  (vz  —  1) 

ac  =  vz$z  (221) 

Then        Oc  =  $1,  the  "fictitious"  primary  flux,  proportional 

to  ii. 
cM  -r-  Oc  =  (vi  -  1) 

OM  =  v&i  (222) 

vi  and  vz  are  here  as  always  Dr.  John  Hopkinson's  stray  coeffici- 
ents. 

Draw  MG  parallel  to  Oa  =  Fz,  to  the  intersection  G  with  the 
extension  of  ON  =  F\. 

Then  the  reader  can  readily  prove  that 

ON 

—  =„  =  Vlv2  -  1  (223) 


222 


INDUCTION  MOTOR 


This  is  our  old  well-known  leakage  coefficient  of  the  transformer. 

It  is  instructive  now  to  draw  the  triangle  LMN  which  is 

taken  from  our  previous  figure.     We  have  chosen  our  scale  for 

Tjl 

this  triangle  arbitrarily  so  that  NM  =  iy  =  — -.  As  all  quanti- 
ties are  entered  into  the  figure,  we  shall  let  the  reader  think  for 
himself. 


FIG.  179. — The   primary   current  locus   of   the   single-phase   induction   motor. 
(ML  perpendicular  to  Oa  See  Fig.  178). 


Draw  LN  to  the  intersection  H  with  the  extension  of  GM 
establishing  at  H.  the  angle  £.  This  angle  we  have  seen  to  be 
constant  and  equal  to 

to-^r 


Thus  H  lies  on  the  arc  of  a  circle  with  NG  as  a  chord. 
Through  M  draw  MK  parallel  to  HN ',  establishing  the  angle 
at  M,  so  that  the  locus  of  M  is  the  circle  over  KG  as  a  chord. 


THE  SINGLE-PHASE  INDUCTION  MOTOR  223 

The  center  C  of  the  circle  KGM  is  readily  seen  to  be  determined 
by  the  angle 

<KGC  =  I  -  £  (224) 

•pi 
Remembering  that   Ey  =  iyr2  and  iy  =  —  and  that   to  ix 

corresponds  the  total' '  fictitious  "  flux  v2$2  in  view  of  the  second- 
ary leakage  path  being  parallel  with  the  air-gap  reluctance,  we 
obtain 

NK      HM      BN 

KG  '"  MG  ~'~  MG 

From  simple  elementary  geometry  the  reader  will  find, 
HM      F»  F» 


MG       v,  '4Fi 

J_Zl 
NO 


I 


HM  -      ff  (<KM 

MG  ~^n 

NG       2(7  +  1 


KG       cr  +  1 

OK       .  <7  + 


ON          2cr  +  1 


(226) 
(227) 


The  last  equation  states  that  the  ratio  of  no-load  current  to 
magnetizing  current  in  the  single-phase  induction  motor  is  equal 
to 


Comparing  this  result  with  that  obtained  in  Chap.  XVII  by 
an  entirely  different  method,  we  find  complete  agreement, 
see  equation  (202).  Also 

2<r  (228) 

The  no-load  current  of  a  single-phase  induction  motor  is  thus 
nearly  twice  the  magnetizing  current  as  the  rotating  field  which 
exists  near  synchronous  speed  has  to  be  supplied  from  one 
phase  instead  of  from  two-  or  three-phases  as  in  the  poly-phase 
motor. 


224 


INDUCTION  MOTOR 


(c)    SUMEC'S  CIRCLES  FOR  SYNCHRONISM,  No  LOAD,  AND 
STANDSTILL 

The  point  P  in  the  previous  figure  moves  to  N  at  synchronous 
speed  S  =  1.  As  PM  is  now  in  time  quadrature  with  the  speed 
field  Fs  the  intersection  with  the  circle  KGM  of  a  semi-circle 
over  NK  at  Ms  marks  the  synchronous  speed  point. 

At  no  load,  which  occurs  slightly  below  synchronism,  we  shall 
prove  presently  that  the  primary  current  lies  at  MI  determined 
as  the  intersection  of  a  semi-circle  over  OK  as  diameter. 


FIG.  180. — The  determination  of  synchronous  speed  and  standstill  in  the  single- 
phase  induction  motor.     (After  Sumec). 

At  standstill  NMG  forms  a  right  angle  as  in  any  stationary 
transformer,  the  vector  of  the  primary  current  being  OMSS. 
These  conditions  are  illustrated  in  the  following  figures.  (Figs. 
180  and  181. 

(d)     THE  INFLUENCE  OF  THE  ROTOR  RESISTANCE  UPON  THE 
PRIMARY  CURRENT  Locus 

In  very  evident  contradistinction  from  the  poly-phase  induc- 
tion motor,  the  primary  current  locus  of  the  single-phase  induc- 
tion motor  depends  very  materially  upon  the  rotor  resistance. 


THE  SINGLE-PHASE  INDUCTION  MOTOR 


225 


It  has  been  seen  that  the  time-phase  angle  between  the  trans- 
former field  and  the  speed  field,  which  we  have  designated  with 
£,  is  a  constant  as  long  as  neither  the  reluctance  of  the  speed 
field  nor  the  resistance  of  the  rotor  undergoes  a  change.  We 
have  seen  that 

(220) 


"7(7 


N\ 


FIG.  181. — The  determination  of  the  no  load  point  on  the  locus  of  the  primary 
current  in  the  single-phase  induction  motor.     (After  Sumec). 

It  is  desirable  that  the  angle  £  approach  7r2  or  that  its  tangent 
approach  infinity.  Hence,  both  the  reluctance  of  the  speed  field 
and  the  resistance  of  the  rotor  should  be  as  small  as  possible. 

As  the  tangent  of  the  angle  KGC  is 


(229) 


15 


226 


INDUCTION  MOTOR 


it  is  directly  proportional  to  the  secondary  resistance  and  the 
distance  of  C  from  KG  is  proportional  to  r2. 

Thus  J.  K.  Sumec  has  drawn  a  diagram  showing  at  a  glance 
the  change  in  primary  current  locus  as  a  function  of  the  resistance 
of  the  rotor,  Fig.  182. 

We  see  at  a  glance,  that,  as  the  center  of  the  circle  rises  from 
0  to  0.2,  the  range  of  the  motor  measured  between  the  "  no-load  " 
and  "standstill"  circles,  is  reduced  to  less  than  one-half.  At 
0.5,  the  motor  can  do  no  work  at  all.  And  above  0.5,  the  energy 
which  is  absorbed  is  dissipated  into  heat. 


FIG.   182. — The  effect  of  resistance  in  the  rotor  upon  the  locus  of  the  primary 
current.     (After  Sumec).     The  single  phase  induction  motor. 

The  range  below  the  abscissa  covers  the  operation  of  the 
induction  machine  as  a  generator. 

(e)  EQUIVALENT  CIRCUITS 

Before  discussing  further  the  properties  of  the  single-phase 
induction  motor,  we  wish  to  outline  briefly  how  a  system  of 
electric  circuits  can  be  built  up  which  will  simulate  the  peculiar 
relations  of  phases  and  load  in  the  single-phase  motor.  While 
these  circuits  appear  more  artificial  here  than  in  the  poly-phase 
motor,  yet  it  is  interesting  to  develop  them. 


THE  SINGLE-PHASE  INDUCTION  MOTOR  227 

Referring  to  Fig.  178  MN  =  Ey  may  be  resolved  into  two 
components,  one  of  which  is  a  watt-component  Mn,  while 
the  other  is  a  watt-less  component  Nn. 

We  thus  obtain  the  following  relations: 


Mn  =  wiF2  - 

=  coiF2(l  -  S2sin2£)  (230) 

Nn  =  coi^2-S2-sin£-  Cos£  (231) 


Two  circuits  must  now  be  substituted  in  which  the  same  cur- 
rents flow,  one  purely  a  watt-current  iw,  the  other  a  watt-less 
current  iwi,  but  these  currents,  passing  through  a  common  react- 
ance £2,  are  fed  from  a  common  voltage  proportional  to  coi/*^- 

Hence,  the  resistance  and  reactance  circuits  must  be  composed 
of  the  following  resistance  and  reactance : 

Mn  =  iwr2  =  coiF2(l  -  S2  sin2  £) 

Also  iw  =  %-2 

/v«> 

where  Rw  is  the  resistance  in  the  equivalent  circuit  to  produce  the 
current  iw  under  an  impressed  voltage  coF2. 


Hence  .- .  ft.  = 

. 


f}  (232) 

Apply  the  same  reasoning  to  the  watt-less  circuit. 
Mn  =  iwi'fz  =  co  \Fz  S2  sin  £  cos  £ 

A  1  •  CO]/'1 2 

Also  iwi  =  -=r- 

where  Xwi  is  the  reactance  to  produce  the  same  watt-less  current 

TT  tr  COlT  2 

Hence  Xwi  =  — — 

=  .8' sin  7  cos{  (233) 

And  <0{=  ^J-  (234) 

(235) 


228 


INDUCTION  MOTOR 


We  thus  obtain  the  following  circuits  which  simulate  the  charac- 
teristics of  the  single-phase  induction  motor,  Fig.  183. 

If  the  reluctance  of  the  speed  field  circuit  is  infinite,  the 
machine  does  not  operate  as  a  motor  as  it  is  permanently  short- 
circuited. 

If  the  reluctance  of  the  speed-field  circuit  is  zero,  then  the 
machine  resembles  somewhat  a  poly-phase  induction  motor  as 
the  extra  watt-less  circuit  is  open  and  sin2  £  =  1. 

Such  a  motor  would  show  only  a  different  speed  characteristic 
from  a  two-phase  induction  motor.  , 

Note. — The  assumption  that  the  effect  of  the  leakage  field  in 
the  rotor  may  be  taken  into  account  by  assuming  that  the  leakage 
of  the  m.m.f.  of  the  currents  of  the  F-system  only  need  be  con- 


32 


5 


Fio.  183. — The  equivalent  circuits  of  the  single-phase  induction  motor  in  the 

cross  flux  theory. 

sidered,  is  an  entirely  arbitrary  one.  It  is  a  very  reasonable 
one  and  it  appears  to  answer  all  practical  considerations  and  to 
lead  to  a  comparatively  lucid  picture  of  the  phenomena  within 
the  machine. 

However,  as  we  have  seen  in  Chap.  IX,  E,  in  discussing  the 
secondary  leakage  reactance  of  the  induction  motor  with  slip 
rings  and  with  commutator,  there  are  here  also  local  leakage 
fields  which  must  induce  e.m.fs.  in  the  conductors  of  the  rotor 
in  such  a  manner  that  the  leakage  fields  produced  by  the  cur- 
rents in  the  speed  field  affect  the  conductors  in  the  transformer 
field. 

This  can  most  readily  be  seen  if  we  study  for  a  moment  the 
conditions  near  synchronism.  At  synchronism  there  exists  a 
nearly  perfect  rotating  field  created  by  the  interaction  of  the 
X-currents  which  are  in  perfect  space  quadrature  and  nearly  in 
time  quadrature  with  the  F-currents. 


THE  SINGLE-PHASE  INDUCTION  MOTOR  229 

That  portion  of  this  field  which  does  not  reach  the  primary, 
cuts  the  rotor  conductors  at  slip  frequency,  thus  the  effect  of  the 
speed-field  currents  consists  in  diminishing  the  leakage  reactance 
of  the  F-system  in  the  rotor. 

(f.)  THEORETICAL  CONSIDERATIONS 

Consider  a  single  turn  in  the  rotor.  Denote  the  flux  at  time  t 
in  the  F-system  with  fy  and  the  flux  at  the  same  time  in  the 
^-system  with  /£ 

ft  =  Fy-sin  co*  (236) 

fi  =  Fx-sm  (co*  -  &  (237) 

Let  the  single  turn  form  the  angle  a.  with  the  F-axis,  then  the 
flux  at  time  t  passing  through  the  coil  is 

ft  =  #cosa+,#sina  (238) 

=  Fy-cos  a-sinco*  +  Fx-sm  a-sin(co*  —  £) 
The  e.m.f.  induced  in  the  coil  at  a  rotor  speed  of  $co  is 
g  dft  da 


dt  dadt 

=  [  —  wFy  cos  co*  —  SuFx  sin  (co*  —  £)]cos  a 
+  [SwFy  sin  co*  —  coFa-  cos  (co*  —  £)]sin  a 
which  may  be  written 

ea  =  ey  cos  a  +  ex  sin  a 
as  we  may  write 

ey  =  —  uFy  cos  co*  —  SuFx  sin  (co*  —  £) 


ex  =  SuFx  sin  co*  -  a>Fx  cos  (co*  -  Q  ' 
The  equations  show  that  the  mode  of  consideration  of  two 
space  fields  in  each  of  which  two  e.m.fs.  are  active,  a  conclusion 
which  we  reached  at  the  outset  of  this  chapter  on  general  physi- 
cal principles,  is  consistent  with  a  more  careful  mathematical 
analysis. 

(g.)  THE  TORQUE 

The  e.m.f.  induced  in  a  single  rotor  turn  at  the  speed  co$  is, 
ea8  =  ~^~  =  (/i-sin  a  -/j-cos  «)£o 

OCX    Ol 

Through  this  conductor  there  flows  a  current 
ia  =  iy-cos  a  +  if'Sm  a. 

1  Throughout  this  paragraph  we  denote  the  "speed  field"  Fs  by  Fx,  and 
F2  by  Ftf. 


230  INDUCTION  MOTOR 

Hence,  the  work  done  upon  this  conductor 

—  ea8i«-dt  =  (fx  cos  a  —  fy  sin  a)(^-cos  a  +  z'^sin  oi)Swdt 

Carrying  out  the  trigonometrical  multiplication  and  summing  up 

the  work  over  the  total  number  of  rotor  conductors  z2,  we  obtain 

dW  =  S(/j  iy  cos2«  -  ft  ix-sm2a)S'wdt 

=  ^(fx  %l  -fv  i£)So>dt, 
and  the  instantaneous  torque, 

"^w*»-*w  (240) 


At  any  moment  the  instantaneous  torque  Tt  consists  of  two 
quantities:  A  positive  torque  produced  by  the  interaction  of  the 
"speed  field"  with  the  "  transformer  currents,"  and  a  negative 
torque  produced  by  the  interaction  of  the  "transformer  field" 
with  the  "  speed  field  currents." 

Let  0  be  the  angle  of  time  lag  of  the  ^/-current  relative  to  the 
main  transformer-field  F2,  then  we  have 

Tt  =  ^z2[Fxiy  sin  (coZ  -  £)-sin  (at  -  6)  - 

Fvix  sin  at-sin  (at  -  (•)]     (241) 
=  ^z2[^'j,{cos  '(0  -  Q  -  cos  (2ut  -  d  -  £)}  - 

Fvif{cos  {  -  cos  (2ut  -  {)}] 

The  resultant  momentary  torque,  composed  of  a  positive  and 
a  negative  part,  pulsates  with  double  frequency  as  each  part 
pulsates  in  this  manner.  The  mean  torque  is  therefore: 

T  =  iz2  [Fxiy  cos  (0  -  &  -  Fvix  cos  £]  (242) 

From  Fig.  178 

Fx  =  SFy-siu  £ 

&»« 

ix  =  —  Fy-cos  £ 

iv  cos  (0  -  Q  =  -Fy  sin  ^(1  -  S2) 

7*2 

By  substitution  in  (242)  above 

T  =  i  ^aFv2S[(l  -  S2)  sin2^  -  cos2^] 
4  TZ 

=  i     -coF,2  S[(2  -  ^2)  sin2?  -  1]  (243) 


THE  SINGLE-PHASE  INDUCTION  MOTOR  231 

These  are  the  equations  as  given  by  Goerges  and  Sumec. 
From   this   equation   follows: 

1.  For  S  =  0  and  for 


=  V2  ~  sln 


the  torque  becomes  zero,  viz.,  at  starting  and  slightly  below 
synchronism.  These  conditions  we  have  already  noted  from  a 
general  consideration  of  the  physical  phenomena. 

2.  Increasing  the  rotor  resistance  diminishes  £  and  therefore  sin 
£  thus  the  torque  decreases  rapidly  as  we  have  seen  clearly  in 
Sumec's  circle  diagram. 

(h)  MECHANICAL  OUTPUT 

From  the  equation  for  the  torque  T  we  have  the  output 
P  =  SuT 


2[S2(l  -  S2)  sin2£  -  S2  cos2£] 

T:    7*2 

The  output  due  to  the  ^/-currents  is  proportional  to,  see  Fig.  178, 

MN-LN-cos  (NML)  =  NP-NL  (244) 

The  energy  loss  per  second  due  to  the  ^-currents  is  equal  to 

i*2T2. 

Ex  =  S-MP 


_ 
~  LP 

MP*  =  PH-PL 
so  that  E2X  =  PH-LN  (245) 

Hence  the  NET  output  is  proportional  to  the  difference  of 
(244)  and  (245), 

P  =  NPLN  -  PH-LN 
=  LN(NP  -  PH) 
=  LN  NQ  (246) 


(i)  ROTOR  COPPER  LOSSES 

In  each  conductor  there  is  dissipated  into  heat  in  time  dt 

rzildt  =  ri(iy  cos  a  +  ilx  sin  a)*dt 
In  summing  up  over  the  circumference  of  the  rotor 

r2S2  cos2*  +    i      sm*a)dt  = 


232  INDUCTION  MOTOR 

Or,  by  integration,  per  unit  time 


Again  from  our  figure 

rstv2  =  PM*  +  PN2 

2[(1  -  >S2)2  sin2^  +  cos2fl  (247) 


Hence  the  total  rotor  loss 

2[(l  -  S2)2  sin2^  +  (1  +  >S2)  cos2^]          (248) 


CHAPTER  XIX 

THE  SINGLE-PHASE  REPULSION  MOTOR 

A.  THE  NON-COMPENSATED  REPULSION  MOTOR 

The  theory  of  the  single-phase  repulsion  motor  can  be  ap- 
proached successfully  in  the  same  way  as  we  have  treated  the 
poly-phase  and  single-phase  induction  motors.  In  fact,  the  use 
of  the  leakage  fluxes  shows  their  great  effectiveness  in  their  ap- 
plication to  these  interesting  motors. 

Throughout  the  treatment  of  the  theory  of  alternating-current 
commutator  machines  it  must  be  borne  in  mind  that  the  phe- 
nomena occurring  in  the  short-circuited  coils  under  the  brushes 
vitiate  to  a  great  extent  all  theoretical  considerations.  Any 
attempt  to  take  the  effects  of  short-circuit  into  account 
proves  disastrous  as  the  complications  that  have  to  be 
introduced  into  the  theory  befog  utterly  the  mind  that  desires 
to  obtain  a  general  understanding  of  the  characteristics  of  these 
motors. 

We  intend  to  limit  our  discussion  to  an  ideal  hypothetical 
motor  without  core  losses  and  without  losses  in  the  coils  under 
the  brushes  or  other  unpleasant  confusing  elements  introduced 
by  the  process  of  commutation.  However,  the  performance  of 
this  ideal  motor  is  very  instructive  and  it  offers  a  clue  to  the 
understanding,  and  a  good  sign  post  to  the  designer,  of  the  real 
motor. 

(a)  The  General  Theory. — A  treatment  of  the  theory  of  the 
repulsion  motor  consistent  with  the  general  theory  which  we 
have  followed  in  this  book,  was  given  almost  simultaneously  by 
M.  Osnos1  and  Andre  Blondel.2  Both  papers  are  beautiful 

1  M.  OSNOS,  E.  T.  Z.,  Oct.  29,  1903. 

2  A.  BLONDEL,  L'Eclairage  Electrique,  Dec.  12  and  26,  1903. 

233 


234 


INDUCTION  MOTOR 


contributions  to  the  simple  elucidation  of  the  complex  phenomena 
in  the  ideal  repulsion  motor. 

Figure  184  shows  diagrammatically  the  circuits  of  the  repul- 


FIG.  184. — Single-phase  repulsion  motor  and  diagram  of  space  phases. 

sion  commutator  induction  motor.  The  axis  Y-Y  is  the  axis 
of  the  stator  field  in  space,  while  B-B  is  the  axis  of  the  short- 
circuited  brushes  on  the  commutator  of  the  rotor. 

At  the  outset,  we  warn  the  reader 
that  he  should  distinguish  carefully 
in  his  mind  between  space-phases 
and  time-phases. 

The  only  way  in  which  the  short- 
circuited  rotor  can  react  upon  the 
stator  is  by  means  of  a  field  whose 
space  axis  lies  in  the  Y-Y  axis. 
Hence  any  rotor  field  reacting  upon 
the  stator  is  going  to  appear  multi- 
plied by  cos  a,  the  cosine  of  the 
angle  of  brush  shift. 

Again,  the  total  field  in  the  stator, 
appearing  in  the  axis  Y-Y,  is  not 
able  to  react  upon  the  rotor.  Only 
such  part  as  falls  into  the  brush 
axis  B-B  can  affect  the  rotor. 

Let,  as  before,  v&i  be  the  total 
"fictitious"  primary  field,  including 
the  primary  leakage  fields,  due  to  the  primary  m.m.f.  Then, 
the  interaction  between  v&i  and  $2  cos  a  from  the  rotor  must 


FIG.  185. — The  flux  diagram  of 
the  single-phase  repulsion  motor. 
(All  time-phases.) 


THE  SINGLE-PHASE  REPULSION  MOTOR  235 

produce,  or  leave  over,   so  to  speak,   the  real  field  FI  which 
balances  the  impressed  primary  voltage. 
Now,  all  time-phases,  Fig.  185. 

OB  =  v&i 
AB  =  $2  cos  a 
OA  =  Fl 

In  order  to  find  the  rotor  fluxes,  we  have  to  consider  that  the 
interaction  of  v&i  cos  a  and  v^z  leaves  over  the  resultant  rotor 
flux  F2  =  OD. 

These  relations  are  those  discussed  again  and  again  in  this 
volume  slightly  altered  quantitatively  only  by  the  mechanical 
configuration  of  the  interacting  systems. 

As  in  the  previous  chapter  we  considered  the  resultant  voltage 
produced  by  the  action  of  the  "transformer"  field  and  of  the 
" speed"  field,  so  we  consider  here  the  similar  e.m.fs. 

The  resultant  "transformer"  field  in  the  axis  B-B  is  F2  which, 
through  the  periodic  rate  of  change,  induces  a  transformer 
voltage  in  time  quadrature  with  F2.  This  voltage  we  shall 
designate  with  eF2T  to  indicate,  as  in  the  cross-flux  theory  of  the 
single-phase  induction  motor,  that  it  is  induced  by  F%  and  by  the 
same  process  as  that  of  transformation. 

At  right  angles  in  space  to  the  brush  axis  B-B  there  exists  a 
"speed"  field  Fs  =  v&i  sin  a  whose  time-phase  coincides  with 
the  time-phase  of  3>i.  Through  rotation  at  angular  velocity  &oi, 

where  S  is  again  the  speed,  S  =  — ,  there  is  induced  a  "speed" 

0)2 

e.m.f .  eFfs  =  SuiFa  =  Suiv&i  sin  a,  which  if  composed  vectorially 
with  ep2T  gives  a  resultant  which  must  be  equal  to  the  ohmic 
drop  in  the  rotor  all  leakages  having  been  taken  into  account 
through  their  respective  leakage  fields.  (Fig.  186.) 

It  is  now  evident,  and  we  remind  the  reader  of  the  procedure 
in  the  previous  chapter,  that  the  angle  at  K  which  is  equal  to 
90  —  £,  is  constant  as  DK :  OD  is  constant. 

Hence  the  angle  at  L  which  is  equal  to  180  —  £  is  constant. 

Draw  O'D  parallel  to  OA,  then  O'L  is  proportional  to  OB  and 
therefore  to  ii,  the  primary  current.  As  L  lies  on  the  arc  O'LG 
vith  C  as  center,  the  primary  current  locus  is  the  arc  O'LG. 

0'L:LD::OB:AB 

v&i:$2  cos  a 


236 


INDUCTION  MOTOR 


r\u 

.   .  U  L  =  v\v<2, 


COS  a 


cos  a 


COS  a 
2  cos  a 

2  cos  a 


FIG.   186. — The    diagram    of    the    single-phase    repulsion    motor. 
NOTE. — In  order  not  to  lose  in  simplicty  of  the  diagram,  we  have  assumed  rz  =  1, 
so  that  the  scale  for  eFsS  and  eP2T  and  LD  could  be  made  the  same. 


O'L  =  t, 
LD  = 


cos  a 


(249) 
(250) 


THE  SINGLE-PHASE  REPULSION  MOTOR  237 

From  simple  proportionalities  follows 

§£  •  (J  -  T?)  (251) 

U  (JT        \  ViVz  I 


Also  =  -  1  (252) 

DG       cos2« 

Hence,  the  leakage  coefficient  of  the  repulsion  motor  is  similar 
to  that  of  the  standard  induction  motor  excepting  that  it  depends 
also  upon  the  brush-shift  angle  a. 

If  a  =  0,  then 

O'D 


as  is  to  be  expected. 

(b)  The  Speed  in  the  Diagram.  —  We  have  found  that  izrz  is 
the  vector  resultant  of  eFaS  and  eFzT- 

eps  —  LK  =  Sui&i  sin  a 


Now,  $> 

i  cos  a-sin  £  =  MD 

Also 

LK-cos  £  =  LM 

T-ToTl  /"»O 

S  co             LjK- 

-LlcilCc 

3>i  sin  a 

LM 

nna   fc. 

MD 

sin  /v 

cos  a  sin 
frcot  «  (253) 

The  speed  of  the  motor  is  therefore  equal  to  the  tangent  of  the 
angle  Z  LDM . 

This  is  zero  at  the  point  M,  hence  Ms  is  the  standstill  point. 

It  approaches  a  maximum,  but  not  infinity,  as  the  primary 
current  decreases. 

(c)  The  Torque. — The  torque  is  proportional  to  the  product 
of  the  "speed"  field  Fs  into  the  rotor  current  component  in  time- 
phase  with  the  time-phase  of  the  "speed"  field,  this  latter  being 
equal  to  the  time-phase  of  the  primary  current,  neglecting  hys- 
teretic  lag. 

Hence,  we  may  write 

T  =  Const. sin  a  LD  — ^—  cos  DLK 

Vi  cos  a 

=  O'LLDtgacos  DLK 


238  INDUCTION   MOTOR 

Construct  a  semi-circle  over  O'D  as  diameter,  then 

LN  =  LD  cos  DLK 
Hence, 

T  =  O'LLN-tga  (254) 

M.  Osnos,  to  whom  this  relation  was  first  due,  Blondel  and 
myself  having  borrowed  it  from  him,  points  out  that,  for  a  given 
primary  current  O'L,  the  rapid  decrease  in  torque  is  due  rather 
to  the  increased  phase  lag  between  the  rotor  current  and  the 
primary  current,  than  to  the  decrease  in  the  rotor  current  itself. 

The  torque,  as  determined  here,  vanishes  at  the  point  P,  hence 
the  maximum  speed  the  motor  is  capable  of  obtaining,  barring 
losses,  is  reached  at  this  point.  The  repulsion  motor,  therefore, 
does  not  run  away  like  the  series  motor. 

(d)  The  Effect  of  the  Rotor  Resistance  upon  the  Diagram. — 
From  the  physical  relations  and  the  diagram  we  have 

0  =  =  P-r2.  Const, 


where  t'J  is  the  magnetizing  current  of  the  rotor  producing  the 
field  F^t  and  p  the  reluctance  of  the  air-gap  path. 

Hence,  the  distance  of  the  center  C  from  the  abscissa  O'G 
is  a  measure  of,  and  proportional  to,  the  rotor  resistance. 

This  result  is  similar  to  that  obtained  in  the  previous  chapter 
in  the  theory  of  the  single-phase  induction  motor. 

Thus  a  number  of  circles  may  be  drawn,  as  in  Fig.  187,  show- 
ing that  the  primary  current  locus  of  the  repulsion  motor  is  a 
semi-circle  only  if  the  rotor  resistance  is  zero.  For  an  infinite 
rotor  resistance,  the  locus  is  a  straight  line  and  no  power  can  be 
developed  by  the  machine.  For  resistances  between  zero  and 
infinity,  the  loci  curves  are  arcs  of  circles  whose  centers  lie  on 
the  same  vertical  line  CCr.  Rotor  resistance  control  can  be 
used  for  speed  regulation  as  the  diagram  indicates.  The  maxi- 
mum speed  and  the  starting  torque  and  starting  current  can  be 
regulated  in  the  same  manner. 

(e)  The  Effect  of  the  Brush  Shift.— Rocking  the  brushes  and 
changing  the  angle  a  changes  the  ratio  of  the  magnetizing  current 
to  the  length  of  the  chord  of  the  arc  of  the  locus  of  the  primary 
current.  It  does  not  affect  the  angle  £. 

We  have  seen  that 

O'D        vivt 
DG    :~cos2a 


THE  SINGLE-PHASE  REPULSION  MOTOR 


230 


O'--r2=8 


FIG.  187. — The  influence  of  rotor  resistance  upon  the  primary  current  locus  of  the 
repulsion  motor.     Fixed  brush  position.     (After  Osnos). 


FIG.  188. — The  influence  of  the  brush  shift  on  the  primary  current  locus  of  the 
repulsion  motor  fixed  rotor  resistance. 


240  INDUCTION  MOTOR 

This  expression  is  a  minimum,  and  therefore  DG  a  maximum 
the  smaller  the  angle  a  and  the  nearer  cos  a  =  1. 

For  a  =  0,  we  obtain  for  this  ratio  ViV2  —  1. 

For  a  =  -,  we  obtain  for  this  ratio  0. 

2i 

Figure  188  shows  these  circles  very  much  as  given  in  the 
brilliant  paper  by  M.  Osnos  already  repeatedly  referred  to. 

The  points  Ms  marking  the  starting  point  lie  on  a  circle  as 
indicated.  The  diagram  suggests  the  mode  of  regulating  the 
speed  and  torque  by  means  of  mechanical  brush-shift  used  at  one 
time  extensively  by  the  Brown  Boveri  Company  for  railway 
motors. 

(f)  Commutation. — The  repulsion  motor  develops  an  elliptical 
rotating  field  as  it  gains  in  speed.  At  or  near  synchronous  speed 
this  field  rotates  with  little  or  no  slip  relative  to  the  rotor,  hence 
the  injurious  effect  of  commutation  through  short-circuit  currents 
induced  in  the  coils  under  the  brush  is  less  in  these  motors  than 
in  motors  of  the  series  type. 

However,  in  starting,  there  is  no  advantage  in  the  repulsion 
motor  over  the  series  motor  and  it  is  in  starting  that  the  trans- 
former effects  in  the  coils  under  the  brush  are  most  injurious. 
It  is  doubtless  on  this  account  that  the  career  of  the  repulsion 
motor  as  a  railway  motor  has  been  rather  brief. 

A  performance  curve  of  an  actual  motor  will  be  given  in  the 
Chapter  on  Commutator  Motors. 


B.    THE    COMPENSATED   REPULSION    MOTOR   OF   WIGHTMAN, 
LATOUR,  AND  WINTER-EICHBERG 

(a)  While  the  compensated  type  of  repulsion  motor  in  which 
the  "speed"  field  is  produced  by  passing  the  primary  current 
through  the  rotor  by  means  of  an  additional  set  of  brushes,  and 
leaving  off  the  field  coil,  has  ceased  to  be  of  practical  importance, 
it  still  offers  an  object  of  great  interest  from  the  point  of  view  of 
its  theory. 

It  should  be  entirely  clear  what  this  motor  really  is,  but  to 
avoid  misunderstanding  let  the  two  types  of  repulsion  motor 
be  placed  side  by  side. 

Supposing  we  wish  to  eliminate  the  field  coil  F  in  the  connec- 
tion diagram,  Fig.  189,  of  the  ordinary  repulsion  motor  in  order 


THE  SINGLE-PHASE  REPULSION  MOTOR 


241 


to  get  rid  of  the  self -inductive  effect  which  causes  a  voltage  drop 
and  a  lowering  of  the  power  factor  of  the  motor.  The  idea 
immediately  suggests  itself  to  utilize  the  armature  itself  for  the 
purpose  of  exciting  the  " speed"  field  and  to  conduct  the  primary 


FIG.   189. — Repulsion  motor  and  compensated  repulsion  motor. 

current  into  the  armature  through  a  set  of  brushes  in  mechanical 
quadrature  with  the  power  brushes. 

It  remains  for  us  to  take  stock  of  the  gain  or  loss  resulting  from 
the  abolition  of  a  field  coil  on  the  stator  and  the  addition  of  an 
extra  set  of  brushes  on  the  rotor. 


X — 


FIG.   190. — Connections    and    space    phases    of    compensated    repulsion    motor. 

Let  us  neglect  leakage  provisionally  in  order  to  get  under  way 
and  to  obtain  a  general  idea  of  the  phenomena  within  the  motor. 

We  make  our  usual  assumption  that  the  active  number  of 
conductors  on  the  stator  and  rotor  is  equal,  z\  =  z^.  Then  we 
have  again  in  space  two  fields,  at  right  angles  to  each  other, 

HI 


242 


INTRODUCTION  MOTOR 


one  being  the  "  transformer  "  field  F  in  the  F-axis,  the  other  being 
the  " speed"  field  F8  in  the  X-axis.  Neither  can  react  upon  the 
other.  (Fig.  190.) 

It  follows  that  the  resultant  magnetization  in  the  F-axis  which 
is  due  to  the  m.m.f.  of  the  stator  iiZi  and  the  opposing  m.m.f. 
i&z  of  the  rotor,  must  produce  a  field  which  balances  the  part  of 

the  impressed  voltage  be- 
tween the  points  A  and  B. 
There  must  be  added  to 
AB  the  voltage  drop  BC, 
in  order  to  obtain  AC, 
which  is  the  total  impressed 
voltage.  Thus  AH  is  the 
primary  m.m.f.  and  HM 
the  secondary  m.m.f.  result- 
ing in  AM  the  magnetizing 
m.m.f. 

However,  the  current  cre- 
ating the  "speed"  field  in 
the  X-axis  induces  through 
transformation  an  e.m.f. 
coiFs  which  lags  in  the  time- 
phase  diagram  Fig.  191  a 
quarter-phase  behind  the 
field.  Let  DG  be  this  e.m.f. 
When  the  motor  stands 
still  we  have  to  add  the 
e.m.f.  DG  with  sign  reversed 
to  AB,  obtaining  BC'  the 
total  impressed  e.m.f. 

Now,  assume  the  rotor  to 

turn  as  a  result  of  the  torque  produced  between  the  current  iz 
and  the  "speed"  field  Fs 

There  will  be  induced  in  the  rotor  four  e.m.fs.  acting  in  two 
pairs,  in  mechanical  space  quadrature. 

First,  as  in  the  case  of  the  repulsion  motor,  there  are  two 
e.m.fs.  set  up  in  the  rotor  in  the  F-axis.  The  e.m.f.  of  trans- 
formation proportional  to  u\F  and  independent  of  the  speed,  and 
the  speed  e.m.f.  Su\Fs  due  to  the  rotor  cutting  through  the 
"speed"  fluxFs  produced  by  the  primary  current  flowing  through 
the  exciting  brushes  in  the  X-axis.  The  resultant  e.m.f. 


FIG.   191. — Time-phase  diagram  of  compen- 
sated repulsion  motor.      (No  leakage.) 


THE  SINGLE-PHASE  REPULSION  MOTOR 


243 


must  fall  in  the  direction  HM  and,  if  we  assume  r2  =  1,  then  HM 
is  a  measure  of  this  e.m.f. 

Secondly,  between  the  exciting  brushes  in  the  rotor  in  the  X- 
axis  there  is  set  up  by  rotation  in  the  "transformer"  field  F  a 
"speed"  e.m.f.  proportional  to  and  in  time-phase  with  F.  The 
direction  of  this  e.m.f. 
must  follow  Lenz's  law 
and  be  opposed  to  the 
transformer  e.m.f.  DG. 

Let  GL  =  Sc^F  be 
this  "speed"  e.m.f., 
then  DL  is  the  result- 
ant e.m.f.  which  must 
be  made  up  by  an 
equal  and  opposite 
e.m.f.  BC  so  that  AC 
is  the  total  impressed 
e.m.f.  upon  the  motor. 

Now,  following  a  sug- 
gestion made  by  M. 
Osnos,1  we  may  intro- 
duce a  hypothetical  or 
imaginary  "  transfor- 
mer" field,  to  the  rate 
of  change  of  which 
there  is  due  the  "real 
speed"  e.m.f.  GL  which 
is  produced  through 
rotation  in  the  field  F. 
(Fig.  192.)  SUIF 

As   We   wish    to    Sim-   FIG.    192. — Time-phase  diagram   of   compensated 

motor.     (No  leakage.)    (After  Osnos.) 


or 


$0=5* 


tffMAN«tgy-S 

N 


Ulate    a   field    <J>0    which 

would  induce,  if  it  existed,  by  transformation  the  e.m.f. 
we  have 

=  coi<J>0 

$0  =  SF 


Hence,  make  MN  =  $o  =  SF,  so  that  the  tangent  of  angle 
MAN  =  tgy  =  S,  then  in  the  force  polygon  DNMA  we  find 
DN  as  the  resultant  field  which  we  have  usually  designated 

1  E.  T.  Z.,  November  12,  1903. 


244 


INDUCTION  MOTOR 


with  FI  and  with  which  the  impressed  e.m.f.  A  C  is  in  time 
quadrature. 

It  is  important  to  note  that,  if  z\  is  not  equal  to  22,  as  would  be 
the  rule  in  an  actual  motor,  then,  as  AD  represented  the  "speed" 
field  in  time-phase  the  " speed"  voltage  being  measured  by  its 
magnitude,  with,  for  instance,  22  =  \z\,  the  speed  voltage  would 
be  S<*>iFs-%Zi}  and  F8  being  itself  proportional  to  22,  it  follows 


R---—-L 


FIG.   193. — Polar,    or    time-phase,    diagram    of    compensated    repulsion    motor. 

(Including  leakage.) 

that  the  "speed"  voltage  is  proportional  to  (-)  .    Hence,  AD 

must  be  multiplied  with  this  ratio  in  order  that  it  can  be  com- 
posed with  AM  and  with  the  hypothetical  field  MN  into  one 
resultant  DN  in  time  quadrature  with  e\. 

(b)  The  Torque. — The  motor  has  two  torques,  a  positive  and 
a  negative  one. 


THE  SINGLE-PHASE  INDUCTION  MOTOR  245 

The  positive  torque  is 

Tl  =  DA  HM  cos  d  (256) 

The  negative  torque  is 

T2  =  -DA- AM  cost  (257) 

as  is  readily  seen  from  Fig.  192.  As  the  speed  increases,  these 
torques  become  more  and  more  equal  to  one  another.  The 
motor  cannot  run  away  and  it  is  in  this  respect  like  the  repulsion 
motor. 

(c)  Performance. — As  will  be  shown  from  an  actual  perform- 
ance curve  in  Chap.  XX,  the  motor  acts  like  a  repulsion  motor 
with  the  same  advantages  as  to  commutation,  approaching  a 
rotating  field  at  speeds  near  synchronism.     Its  power  factor 
increases  with  the  speed. 

(d)  Leakage. — The  leakage  is  taken  into  account  in  Fig.  193. 
As  usual  we  lay  down  the  total  primary  " fictitious"  flux  which 
would  be  produced  if  the  primary  m.m.f.  acted  alone  on  the 
magnetic  circuit.     This  is  AH'  =  ViAH. 

EM'  =  v2HM 
AM  =  F\ 

AM'  =  F2 

M'N'  =  <J>o  =  SF* 

DN  =  FI  which  is  not  a  real  field,  the 

real  primary  field  in  the  F-axis  being  AM  =  F\,  DN  =  FI  being 
a  hopothetical  field  as  the  "speed"  voltage  induced  by  rotation 
in  the  field  F%  is  not  represented  by  a  field  in  the  7-axis.  To 
obtain  the  real  physical  conception  of  the  phenomena  in  the 
motor,  it  is  well  to  adhere  to  Figs.  191  and  192. 


CHAPTER  XX 
SINGLE-PHASE  COMMUTATOR  MOTORS 

A  CONDENSED  REVIEW 

With  the  appearance  in  1902  of  Mr.  B.  G.  Lamme's  paper  on 
the  application  of  single-phase  commutator  motors  to  railway 
work,1  a  new  impetus  was  given  to  the  inventors  and  engineers 
the  like  of  which  had  not  been  witnessed  since  Tesla's  great 
invention.  Though  the  particular  installation  referred  to  in  the 
paper  was  never  actually  executed,  though  the  motor  was,  in 
Steinmetz's2  expression,  "our  old  friend"  the  single-phase  com- 
mutator motor,  yet  there  was  infused  new  hope  and  energy  into 
the  railway  field.  The  feverish  activity  which  absorbed  the 
engineering  community  for  10  years  after  the  reading  of  Mr. 
Lamme's  paper  resulted  in  the  creation  of  innumerable  types 
which,  while  they  varied  only  slightly  in  the  manner  of  their 
operation,  attracted  attention  altogether  beyond  their  intrinsic 
value  and  interest. 

Nineteen  years  have  passed  and  a  more  sober  frame  of  mind 
has  superseded  the  fond  dreams  of  that  early  period.  A  few 
types  have  survived  among  them  particularly  the  conductively 
compensated  series  motor  with  interpoles  in  shunt  connection 
and  the  repulsion  induction  motor  already  treated  in  Chap.  XIX. 

1  B.    G.    LAMME,    "Washington,    Baltimore    &    Annapolis    Single-phase 
Railway,"  Trans.  A.  I.  E.  E.,  September,  1902. 

2  "I  believe  we  can  congratulate  ourselves  then  that  here  is  published 
the  record  of  some  work  done  in  the  direction  of  developing  apparatus, 
giving   the   proper   characteristics  for   alternating    current   railway   work. 
I  must  confess,  however,  that  I  have  been  somewhat  disappointed  in  reading 
this  paper,  by  seeing  that  after  all  this  new  motor  is  nothing  but  our  old 
friend  the  continuous  current  series  motor  adapted  to  alternating  currents 
by  laminating  the  field.     Now,  I  remember  this  type  of  motor  very  well 
because  I  was  associated  with  Mr.  EICKEMEYER  in  1891  and  1892,  and  we 
spent  a  very  great  deal  of  time  in  building  alternating  current  series  motors, 
investigating  their  behavior,   and  trying  to  cure  them  of  their  inherent 
vicious   defects."     MR.    C.    P.    STEINMETZ'S   discussion  of   MR,   LAMME'S 
paper,  Trans.,  A.  L  E.  E.,  Sept.  6,  1902. 

246 


(Facing  page  246) 


SINGLE-PHASE  COMMUTATOR  MOTOR  247 

A.  VARIETY  OF  TYPES  OF  SERIES  A.  C.  COMMUTATOR  MOTORS 

In  the  accompanying  diagrams  a  number  of  connections  are 
shown  representing  a  few  of  the  very  large  number  of  schemes 
which  have  been  brought  to  light  as  a  result  of  the  feverish  in- 
ventive activities  in  this  direction. 

Many  of  these  connections  are  quite  old  and  interest  in  them 
was  revived  when  single-phase  motors  again  commanded  atten- 
tion as  a  result  of  the  work  of  Mr.  Lamme  and  Mr.  Westinghouse. 

For  instance,  the  interesting  compensated  motor  later  re- 
invented by  Marius  Latour  and  Winter  and  Eichberg,  is  described 
fully  in  the  U.  S.  patent  by  M.  J.  Wightman,  No.  476,  346, 
dated  June  7,  1892,  and  assigned  to  the  Thomson-Houston  Elec- 
tric Company. 

Mr.  Lamme's  first  motors  were  not  compensated.  They  had 
a  field  winding  of  few  turns  and  a  magnetic  frame  of  low  reluct- 
ance. To  handle  the  transformer  current  so  troublesome  in 
starting  and  at  low  speeds,  induced  in  the  coils  short-circuited 
under  the  brushes,  Mr.  Lamme  used  resistance  leads  ingeniously 
embedded  in  the  slots  side  by  side  with  the  conductors.  We 
refer  the  reader  to  U.  S.  patents  No.  758,  667,  May  3,  1904,  and 
to  No.  758,668  of  the  same  date. 

It  is  very  easy  and  very  human  and  natural  to  remark,  as  was 
done  at  the  time  when  Mr.  Lamme  read  his  epoch-marking  paper 
that  there  was  little  that  was  new  in  the  system  he  described. 
That  statement  may  be  admitted.  Yet  no  one  had  used  these 
old  ideas  and  no  one  had  designed  a  workable  single-phase  com- 
mutator motor.  I  think  the  present  author  may  claim  fairly 
for  himself  that  he  never  became  a  single-phase  enthusiast.  The 
limitations  of  the  system  were  too  deeply  forced  upon  his 
attention  in  his  work  done  in  the  middle  nineties.  At  no  time 
did  he  share  the  enthusiasm  expressed,  for  instance,  by  Mr. 
Steinmetz  in  the  following  telling  passage:1 

"If  I  may  be  permitted  to  take  a  look  into  the  future,  although  we 
do  not  know  what  to-morrow  will  bring,  I  think  the  system  of  the  future 
will  be  the  single-phase  system.  Where  the  power  is  transmitted  over 
a  long  distance  by  an  overhead  wire,  the  ground  can  be  used  as  the  return 
conductor.  .  .  .  But  if  we  use  a  single-phase  current  in  the  power 
transmission  of  the  future,  then  we  will  have  to  learn  many  things." 

1  "Proceedings  of  the  International  Electrical  Congress  held  in  the  City 
of  Chicago.  Published  by  the  A.  I.  E.  E.,  New  York,  1894,  p.  437. 


248  INDUCTION  MOTOR 

The  present  writer  sounded  a  note  of  scepticism  and  caution 
in  a  paper  in  Gassier' s  Magazine,  May,  1907,  where  he  said : 

" There  is,  in  the  realm  of  ideas,  a  distinct  difference  between  'nat- 
ural' ideas  and  'forced'  ideas.  The  natural  idea  may  be  likened  to  a 
plant  growing  under  favorable  conditions  and  adapting  itself  to  its 
environment;  the  forced  idea  may  be  likened  to  a  plant  raised  in  a  hot- 
house, with  the  exclusion  of  such  conditions  as  might  have  a  tendency 
to  prevent  its  development.  The  natural  idea  will  survive;  the  forced 
idea  will  go  to  the  wall ;  but  it  is  often  only  after  extended  experiments 
conducted  on  a  large  scale  have  been  laboriously  completed  that  we 
realize  that  an  idea  has  been  followed  out  which  could  have  lived  only 
under  particularly  favorable  conditions,  such  as  are  not  usually  found 
in  practical  operation.  In  contemplating  the  history  of  the  develop- 
ment of  the  utilization  of  alternating  currents,  the  single-phase  system 
has  appeared  to  be  an  almost  ideally  simple  system.  It  is  only  too 
obvious  that,  if  power  could  be  safely  transmitted  and  utilized  in  an 
economical  manner,  and  by  means  of  simple  mechanical  apparatus,  the 
generation  and  utilization  of  single-phase  currents  would  soon  replace 
the  poly-phase  system.  Such  attempts  were  made  15  years  ago,  and, 
after  considerable  effort,  most  engineers  abandoned  hope  in  developing 
a  practical  system  of  transmission  of  energy  by  means  of  single-phase 
currents.  The  experience  gained  during  the  past  15  years  with  -poly- 
phase currents  and  the  many  opportunities  afforded  the  engineer  for 
comparing  the  single-phase  machinery,  generators,  and  motors,  with 
their  poly-phase  cousins,  have  led  to  an  attitude  of  skepticism  towards 
single-phase  current.  The  very  much  reduced  output  of  both  generators 
and  motors  if  operated  single-phase;  the  reduced  efficiency;  the  impaired 
regulation;  the  increased  heating,  and  the  lesser  stability  of  single-phase 
motors  and  generators,  connected  with  the  increased  cost  as  produced 
by  the  greater  amount  of  material  required ;  these  form  the  main  reasons 
for  inducing  me  to  call  the  recent  attempts  which  have  been  made  in 
the  utilization  of  single-phase  current,  a  forced  idea." 

And  later  in  the  same  paper  the  present  author  remarked : 

"  Those  gigantic  experiments  to  be  conducted  on  the  New  York,  New 
Haven  &  Hartford  Railroad  are  being  watched  with  the  respect  due  an 
enterprise  of  such  magnitude,  and  with  the  hope  that,  even  if  the  sys- 
tem should  not  be  all  that  its  ingenious  designers  had  expected,  it  may 
yet  lead  on  to  ideas  which  will  finally  solve  the  problem  of  electrically 
operating  the  present  steam  railways  of  the  world." 

Starting  with  the  non-compensated  series  motor  (1)  we  pro- 
ceed to  the  type  with  conductively  compensated  armature  (2) 


SINGLE-PHASE  COMMUTATOR  MOTOR 


249 


and  then  the  inductively  compensated  connection  suggests  itself 
(3).     (Fig.  194.) 

TYPES  OF  SINGLE- PHASE  COMMUTATOR.MOTORS 


(1)  Plain  Series 
Motor 


(2)  Conductively 
Compensated  Series 

Motor 
lEickemeyer,  Lamme) 


( 3)  Inductively 
Compensated  Series 

Motor 
(Eickemeyer) 


(4) Inverted  Repulsion 

Motor,  Series  Motor 

with  Secondary 

Excitation 


(5)  Repulsion  Motor        (6)  Repulsion  Motor     (7)  Series  Repulsion  (8) Series  Repulsion 
with  Stator  E-xcitation    Motor  with  Secondary     Motor,  with  Primary 

(Atkinson)  Excitation  ,A,  ..  Ex,citatio? 

('Doubly-Fed ) 


(9)  Compensated 

Repulsion  Motor 

(Wightman,  Latour,) 

(Winter,Eich^erg) 


(10)  Rotor-Excited 
Series  Motor 

with  Conductive 
Compensation 


(11)  Rotor-Excited 

Series  Motor 
with  Inductive 
Compensation 


(12)  Series  Motor  with 
Rotor  Excitation 
and  Compensation 
(McAllister) 


(13)  Induction  Repulsion 
Motor  (Atkinson) 


(14)  Commutator 

Induction  Motor 

(.Atkinson ) 


(15)  Repulsion  Motor    (16)  Single-Phase  Unity 
(Deri,Latour)  Power  Factor  Motor 

(Wagner-Fynn) 


(17)  Shunt  Motor 
(Creedy) 


(IS)  Shunt  Motor 
(Fynn) 


(19)  Series  Motor  Conductively 

Compensated  with  Interpole 

in  Shunt 


FIG.  194. — Single-phase     commutator     motors.     A     variety     of     connections. 


As  the  next  step  we  may  excite  both  the  field  and  the  com- 
pensating coil  inductively  (4). 

By  exciting  the  field  and  compensating  windings  from  the  bus 


250  INDUCTION  MOTOR 

bars  and  short-circuiting  the  rotor  we  obtain  one  of  the  many 
types1  of  repulsion  induction  motor  (5). 

By  splitting  field  and  compensating  winding,  we  obtain  Figs. 
(6)  and  (7). 

By  feeding  both  the  field  and  compensating  windings  in 
series  from  one  potential  and  the  armature  from  another  we 
obtain  the  interesting  "doubly-fed"  motor  of  Latour2  and 
Alexanderson.3 

By  exciting  the  field,  which  is  located  on  the  stator  in  the 
repulsion  motor  (5),  in  the  rotor  proper,  by  means  of  an  additional 
set  of  brushes  at  right  angles  to  the  short-circuited  brushes, 
we  arrive  at  the  Wightman-Latour-Winter-Eichberg  compen- 
sated motor  (9)  the  chief  advantage  of  which  lies  in  better  com- 
mutation and  higher  power  factor  than  in  the  repulsion  induction 
motor.  The  extra  brushes  are  a  grave  mechanical  drawback 
and  they  would  seem  to  be  too  high  a  price  to  pay  for  the  advan- 
tage obtained. 

(10),  (11),  (12)  the  McAllister  connection,  and  (13)  are 
modifications  of  (9). 

(14)  is  a  single-phase  commutator  induction  motor,  also  due 
to  Atkinson,  I  believe. 

Deri  and  Latour  suggested  (15),  Fynn  suggested  (16),  Greedy 
suggested  (17),  and  (18)  is  again  a  Fynn  motor. 

(19)  appears  to  represent  the  chief  survivor  of  this  great  host 
of  types.  It  is  a  plain  series  conductively  compensated  single- 
phase  commutator  motor  with  interpoles  excited  from  the 
impressed  voltage  so  as  to  produce  a  commutating  field  of  the 
right  time-phase.  If  motors  of  this  type  are  started  with  direct 
current  and  operated  on  single-phase  alternating  current,  then 
commutation  will  be  very  satisfactory. 

B.  OPERATING  CHARACTERISTICS  OF  DIFFERENT  TYPES 

In  order  to  give  a  general  idea  of  the  performance  of  some  of  the 
types  of  single-phase  commutator  motor  which  have  been 
enumerated  here  we  reproduce  from  tests  the  following: 

1  See  LLEWELYN  BIRCHALL  ATKINSON,  Proceedings,  Institution  of  Civil 
Engineers  of  Great  Britain,  Feb.  22,  1898.     "The  Theory,  Design  and  Work- 
ing of  Alternate-Current  Motors." 

2  M.  C.  A.  LATOUR,  U.  S.  Patent  No.  841,257,  Jan.  15,  1907. 

3  E.  F.  W.  ALEXANDERSON,  U.  S.  Patent,  No.  923,754,  June  1,  1909. 


SINGLE-PHASE  COMMUTATOR  MOTOR 


251 


Figure  195  shows  the  characteristics  of  a  repulsion  motor  like 

(5). 


500.  SC 


300- 

200_  20_]40.| 
•4 
100-1 10-20- 

20 


"•Vj<.ot  Single  Phase  R'w'y  Mojtor  I"    'O 
— Repulsion-Motor  hConnection- — I- 


Operating  on  200  Volts',  25 1  Cycles 


Brushea  Shi 


Rotation,  o 


19  4°(Elec.) 


Ampcies  Line 
I      100      j      120 


140 


FIG.  195. — Operating  characteristics  of  a  single-phase  repulsion  motor. 


FIG. 


196. — Operating  characteristics  of  a  single-phase  repulsion  motor.     Latour 
connections. 


Figure  196  shows  the  performance  characteristics  of  a  com- 
pensated   repulsion   motor   with  Latour   connections   like    (9). 


252 


INDUCTION  MOTOR 


Figure  197  shows  the  performance  characteristics  of  the  same 
motor  connected  as  suggested  by  A.  S.  McAllister  in  (12). 


L4U 

1400 

<s^ 

*y 

r 

j-Vc 

Y 

X)_vl 
Its- 

\ 

st 

'Us 

itor 

1200 

\ 

<$ 

P^ 

Ar 

x 

X 

^ 

> 

> 

100- 

"I 

03 

•I 

« 
—  s 

40> 

100- 

"I 

80  £ 

70-| 
60^ 
59-2" 

§ 

401 

20- 

1000 

"fc 

—  pj 
a 
600- 

400- 

20 

X 

-Po 

^y 

\ 

/ 

18 

X 

X 

5 

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>xx 

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—  0— 

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A. 

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s. 

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s# 

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BW 

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cy\ 

^ 

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10 

2 

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N 

V- 

g 

N 

\ 

Single  Phase  Rwy.  Motor 
McAllister  Connection 
Operating  on  100  Volts,  25  AJ 

20- 

200 

Rotation  -  C 
Brushes  within  &°(Elec.) 
of  Neutral  Position 

2 

j 

0 

5 

0 

IS 

B 

1 

50 

2' 

Juper 

ea  Line 
240 

2 

50 

1 

320 

Fia.  197. — Operating    characteristics    of    a    single-phase    commutator    motor. 
McAllister  connections. 


FIG.  198. — Operating  characteristics  of  a  single-phase  commutator  motor  on 

direct  current. 


Figure  198  shows  the  performance  of  the  same  motor  operating 
as  a  direct  current  motor. 


SINGLE-PHASE  COMMUTATOR  MOTOR 


253 


Examination  and  comparison  of  these  curves  are  left  to  the 
reader. 

Figure  199  shows  the  performance  of  one  of  the  latest  types 
according  to  (19)  designed  by  the  Westinghouse  Company. 


I  To  Line  , 

Transformer 
^v^/^AA^/^xv^/SA^^/^/^A^/^/V^^ 


400  4000 


300  3000 


200  2000 


-  100  1000 


1000 


1200 


600  800 

Line  Amperes 
Note: -Above  curves  are  based  on  Hue  amperes  assuming  perfect 

transformer  (no  impedance   or  exciting  current)  and  1:1  ratio. 

FIQ.   199. — Operating  characteristics  of  conductively  compensated  series  com- 
mutator motor.     One  of  the  latest  types  of  the  Westinghouse  Company. 


C.  METHODS  OF  IMPROVING   COMMUTATION' 

(a)  Resistance  Leads  and  Limits  of  Voltage  for  Commutation. 

The  re-introduction  by  Mr.  Lamme  of  resistance  leads,  already 
known  in  connection  with  the  development  of  the  direct-current 
generator,  to  take  care  of  the  transformer  current  under  the 
brushes,  appeared  at  first  as  a  happy  solution  of  a  very  trouble- 
some problem.  In  spite  of  the  ingenious  mechanical  disposition 
of  these  resistance  leads,  they  became  a  source  of  great  trouble 

1  See  a  valuable  paper  by  B.  G.  LAMME,  "The  Alternating-Current 
Commutator  Motor,"  Journal,  A.  I.  E.  E.,  April,  1920.  Also,  E.  ARNOLD, 
Vol.  V,  Part  II. 


254  INDUCTION  MOTOR 

on  account  of  mechanical  break-down,  doubtless  due  to  vibration 
difficult  to  cope  with.  These  resistance  leads  have  been 
abandoned. 

In  the  paper  referred  to  below,  Mr.  Lamme  gives  as  a  prac- 
ticable average  voltage  under  the  brush,  which  will  still  permit 
satisfactory  operation  in  railway  motors  operating  under  average 
conditions  not  exceeding  one  hour  full  load,  a  maximum  of  10  to 
12  volts.  No  general  rules  can  be  laid  down. 

(b)  (1)  The  object  of  the  interpole  in  a  direct-current  machine 
consists  in  generating  an  e.m.f.  in  the  coil  under  commutation 
equal  and  opposite  to  the  e.m.f.  of  self-induction  which  tends  to 
prevent  the  rapid  rate  of  change  of  direction  necessary  for 
sparkless  commutation. 

The  same  object  is  obtained  in  the  alternating-current  series 
motor  by  a  series  coil  placed  upon  an  interpole  so  that  the 
interpole  flux  is  in  time-phase  with  the  "speed"  field  and  hence 
with  the  primary  current. 

(2).  In  addition  to  the  creation  of  an  e.m.f.  which  takes  care  of 
the  e.m.f.  of  self-induction,  it  is  necessary  to  compensate  for  the 
e.m.f.  induced  through  transformation  in  the  coil  under  commuta- 
tion. As  this  e.m.f.  has  to  be  produced  by  rotation  in  order  to 
have  the  same  phase,  it  must  be  produced  by  a  flux  in  time 
quadrature  with  the  "speed"  field.  However,  the  "speed" 
field  is  in  time-phase — barring  hysteresis — with  the  primary 
current,  hence  the  exciting  current  of  the  interpole  must  be  in 
time  quadrature  with  the  primary  current. 

Dr.  Behn-Eschenburg  and  Latour  shunt  the  interpole  with  a 
non-inductive  resistance  to  obtain  the  right  phase  relation  to 
take  care  of  both  (1)  and  (2). 

As  pointed  out  by  Arnold,  the  time  phase  of  the  transformer 
e.m.f.  can  be  compensated  by  a  connection  of  the  interpole 
to  the  rotor,  while  the  interpole  winding  should  be  con- 
nected to  the  field  coil  whose  voltage  is  in  quadrature  to  the 
current  in  order  that  it  should  compensate  the  e.m.f.  of  self- 
induction.  Thus,  if  the  interpole  coil  is  connected  across  the 
impressed  potential  of  the  motor,  a  good  general  effect  may  be 
obtained. 

As  the  compensation  of  the  "transformer"  voltage  by  means 
of  a  "speed"  voltage  requires  rotation,  compensation  at  starting 
when  it  is  most  needed,  cannot  be  obtained  in  this  manner. 


SINGLE-PHASE  COMMUTATOR  MOTOR 


255 


D.  THE  SHUNT  EXCITED  A.  C.  COMMUTATOR  MOTOR1 

Referring  to  our  compilation  of  diagrams  of  connections  the 
types  (13),  (14),  (15),  (16),  (17)  and  (18)  may  be  considered 
as  shunt  types  in  contradistinction  to  the  series  types. 

A  plain  shunt  connection  like  Fig.  200  cannot  be  effective  as 
the  time-phase  of  the  field  flux  lags  approximately  a  quarter 
of  a  period  behind  the  impressed  e.m.f.  and  therefore  there  can 
be  no  torque  developed  by  the  rotor  as  the  field  flux  and  the 
rotor  current  would  be  in  time  quadrature. 

(a)  It  has  been  suggested 
to  use  condensers  for  ob- 
taining time-phase  equality 
between  the  flux  and  the 
impressed  e.m.f. 


FIG.    200.  —  Single-phase    shunt 
motor.      (Inoperative). 


FIG.  201. — Single-phase  shunt  motor  with 
Behn-Eschenburg  connection. 


(6)  Another  method  perhaps  equally  undesirable  has  been 
suggested  by  Dr.  Behn-Eschenburg,  which  we  wish  to  mention 
here  because  of  its  theoretical  interest.  (Fig.  201.) 

In  series  with  the  exciting  coil  xi,  which  is  to  produce  the  field 
of  the  motor,  there  is  connected  a  resistance  r\  and  the  field  coil 
Xi  and  resistance  n  are  shunted  by  an  impedance  Xzrz  and  in 
series  with  these  parallel  circuits  there  is  connected  a  resistance 
r.  In  this  manner  the  field  current  ii  can  be  brought  in  phase 
with  the  impressed  e.m.f.  E. 

vSee  F.  GREEDY,  "Single-phase  Commutator  Motors."  New  York, 
D.  Van  Nostrand  Co.,  1913. 

E.  ARNOLD,  "Die  Asynchronen  Wechselstrommaschinen,"  Vol.  V, 
Part  II,  by  E.  ARNOLD,  J.  L.  LA  COUR  and  A.  FRAENCKEL.  Berlin, 
J.  SPRINGER,  1912,  p.  484  et  seq. 


256 


INDUCTION  MOTOR 


Let  us  take  the  general  case  of  two  shunted  impedances  with 
a  resistance  in  series.  It  is  interesting  to  investigate  under  what 
conditions  the  current  i\  passing  through  one  of  these  impedances 
is  in  time-phase  with  the  impressed  e.m.f.  applied  at  the  points 
A  and  B. 

The  vector  diagram  for  these  circles  is  easily  drawn  for  the 

condition  to  be  investigated,  viz., 
phase  equality  of  i  and  e\  Figs. 
202  and  203, 


OA 

OCiOA'.iizZz'.iin 


sin  a  =  i  — —  —  l\  — 
IT  r 


.    f\ri  °  _          i 


OB  =  i-2,  cos  7  =  1 
BC:BD::AC:OA 
AC  =  OC-sin  |8  = 


22 


.-.  BD 


B 

FIG.  202. — Combination  of  resist- 
ances and  reactances  to  produce 
desired  time-phases  in  the  field  and 
armature  of  single-phase  shunt  motor. 


BC 


BC  =  i 


OB  -  BC  =  OC 

.  X2 


22 


Hence 


(258) 


For  a:2  =  22,  also  r2  =  0 

Hence  rri  =  xtf*  (259) 

This  beautiful  scheme  proves  impracticable  on  account  of  its 


SINGLE-PHASE  COMMUTATOR  MOTOR 


257 


low  efficiency.  If  we  look  into  this  carefully  we  find  that  under 
the  most  favorable  conditions  the  ratio  of  the  kw.  dissipated 
into  heat  in  the  resistances  to  the  exciting  kva.  is  5.2.  There- 
fore, if  we  assume  that  the  exciting  kva.  equal  about  one-fifth 
of  the  output,  then  one-fifth  of  5.2  is  equal  to  1.04  or  the  kw. 
dissipated  in  the  resistances  to  obtain  phase  equality  between 
the  exciting  current  and  the  impressed  e.m.f.  of  the  motor  are 


FIG.  203. — Time-phase  diagram  for  combination  of  resistances  and  reactances. 


4  per  cent  larger  than  the  entire  output  of  the  motor.  Such  a 
motor,  therefore,  considering  its  other  losses,  would  have  an 
efficiency  of  about  40  per  cent. 

To  prove  this,  form  the  ratio  X  of  kw.   dissipated  to  kva. 
excitation. 


17 


258  INDUCTION  MOTOR 

This  expression  is  a  minimum  for 


and  X  =  -^  =  5.2  Q.E.D. 


E.  THE  SUPPLY  OF  SINGLE-PHASE  POWER  FROM  THREE-PHASE 

SYSTEMS 

With  the  growth  of  certain  single-phase  developments  like 
the  systems  of  the  Pennsylvania  Railroad  between  Broad  Street 
Station,  Philadelphia,  and  Paoli,  and  of  the  New  York,  New 
Haven  &  Hartford  Railroad  between  Woodlawn  and  New 
Haven,  single-phase  power  has  to  be  purchased  from  public 
service  stations  generating  three-phase  currents. 

The  armature  reaction  in  turbo  generators  caused  by  an 
unbalanced  three-phase  system  due  to  the  single-phase  load  is 
very  troublesome.  This  is  so  much  so  that  the  use  of  single- 
phase  substations  has  been  considered  so  as  to  be  able  to  generate 
single-phase  power  in  machines  of  comparatively  moderate 
speed  built  in  such  a  manner  that  the  effect  of  the  double-fre- 
quency m.m.f .  of  the  armature  can  be  dampened  out  of  existence 
by  powerful  damping  circuits  placed  in  the  pole-faces  of  the 
generators.  This  is  comparatively  easy  to  do  in  moderate  speed 
machines,  but  very  difficult  in  turbo-generators.1 

Some  very  ingenious  work  has  been  done  recently  by  E.  F.  W. 
Alexanderson2  and  by  R.  E.  Oilman  and  C.  Le.  G.  Fortescue.3 
A  very  able  survey  has  been  published  by  Prof.  Miles  Walker4 
to  which  we  must  refer  for  want  of  space.  However,  we  wish 
to  copy  from  Prof.  Walker's  paper  the  simple  underlying  princi- 

1  See  A.  B.  FIELD  and  B.  A.  BEHREND,  U.  S.  Patent  No.    1,269,590, 
June  18,  1918. 

2  See  E.  F.  W.  ALEXANDERSON  and  G.  H.  HILL,  "Single-phase  Power 
Production,"  Trans.  A.  L  E.  E.,  Oct.,  1916,  p.  1315. 

3  Ibid.,  p.  1329. 

4  MILES  WALKER,    Transactions.     "The   Institution  of  Electrical  Engi- 
neers," of  Great  Britain,  Nov.,  1918. 


SINGLE-PHASE  COMMUTATOR  MOTOR 


259 


pie  of  the  breaking  up  of  an  elliptically  rotating  field,  produced 
by  unbalanced  poly-phase  currents,  into  two  unequal  oppositely 
rotating  magnetic  fields. 

'  'Consider  the  general  case  of  any  elliptically-rotating  field.  Let  the 
ellipse  be  that  shown  in  Fig.  204.  Whatever  be  the  slope  of  the  major 
axis  with  respect  to  the  horizontal  line,  it  is  always  possible  to  take  our 
abscissae  x  along  the  major  axis  and  our  ordinates  y  along  the  minor 
axis,  and  to  express  the  curve  in  the  form  x2/a2  -\-  ?/2/62  =  1,  where  a 
and  b  are  the  lengths  of  the  major  and  minor  axes  respectively.  If  we 
now  write  a  =  b  -\-  c,  where  c  can  be  found  in  any  given  case, 


and  ^  = 
62 


(6  +  c)2 

x  =  b  cos  ut  +  c-cos 
y  =  b  sin  ut 


The  elliptically-rotating  field  can  be  regarded  as  consisting  of  two  parts: 
one  part  a  simple  rotating  field  whose  vector  is  given  by  the  coordinates 
xi  =  b  cos  ut,  yi  =  b  sin  o>i,  and  the  other  part  a  single-phase  stationary 
field  lying  along  the  major  axis  of  the  ellipse,  given  by  the  expression 


xz  =  c-cos 


=  0 


Now,  the  single-phase  field  can  be  broken  up  into  two  oppositely-rotating 
fields  : 


2/2    =   2/4  +  2/5 


c- sin  (at  + 


260  INDUCTION  MOTOR 

Adding  Xi  to  x*  and  y\  to  2/4,  we  get: 

cos  w* 

sin  w* 

These  form  a  uniform  field  rotating  forwards,  while  x&  =  %c-cos  (  —  cot) 
and  2/5  =  H^'sin  ( —  <*t)  give  us  a  uniform  field  rotating  backwards.  Thus 
we  see  that  the  magnetic  field  produced  in  a  poly-phase  machine  having 
symmetrically  disposed  coils  carrying  an  unbalanced  load  can  be  re- 
garded as  the  resultant  of  two  uniform  fields  of  different  amplitudes 
rotating  in  opposite  direction." 

The  general  principle  enunciated  in  Oilman  &  Fortescue's 
paper,  viz.,  that  "an  unbalanced  poly-phase  system  can  be  re- 
solved into  two  balanced  systems  of  positive  and  negative  phase- 
rotation,"  does  not  appear  to  have  been  proved  rigorously  in  that 
paper.  The  subject  is  of  increasing  importance  and  would  have 
merited  more  space  than  this  passing  reference. 


APPENDIX 

It  became  incumbent  upon  the  writer  more  than  20  years  ago, 
to  appear  as  though  he  gave  countenance  to  the  infringement 
of  the  fundamental  Tesla  patents.  A  large  number  of  induction 
motors  designed  by  him  during  the  life  of  these  patents,  which 
constituted  a  plain  infringement  of  Tesla's  inventions,  have  no 
doubt  been  pointed  to  as  an  indication  that  he  either  did  not 
believe  in  the  validity  of  these  patents  or  that  he  deliberately 
became  a  party  to  their  infringement. 

The  Company  of  which,  at  the  period  referred  to,  he  was  Chief 
Engineer  owed  its  growth  and  development  largely  to  his  personal 
efforts  in  the  design  and  development  of  electrical  machinery  and 
to  his  success  in  organizing  an  effective  engineering  staff,  con- 
sisting of  a  number  of  eminent  men  among  whom  were  David 
Hall,  A.  B.  Field,  W.  L.  Waters,  Bradley  T.  McCormick,  H.  A. 
Burson,  Alexander  Miller  Gray,  R.  B.  Williamson,  Carl  Fech- 
heimer  and  others.  In  due  course  the  owners  of  the  Tesla 
patents  proceeded  against  our  company  and  in  the  long  litigation 
which  followed  the  writer's  position  was  at  times  embarrassing 
and  disagreeable.  By  way  of  epilogue,  he  begs  leave  to  publish 
now,  with  the  bitterness  of  the  controversy  abated,  a  letter 
addressed  to  the  patent  counsel  of  his  Company: 

Cincinnati,  Ohio,  May  23d,  1901. 
MR.  ARTHUR  STEM, 

PATENT  ATTORNEY, 

CITY. 
My  dear  Sir: 

Enclosed  please  find  my  notes  on  the  Record  of  Final  Hearing  in  the  suit 
of  Westinghouse  Electric  &  Mfg.  Co.  vs.  the  New  England  Granite  Co. 

You  will  see  that  I  am  now,  even  more  than  I  have  been  before,  of  the 
opinion  that  it  is  not  possible  for  us  to  bring  forth  arguments  that  could  go 
to  show  the  invalidity  of  the  Tesla  Patents  in  suit.  While  I  am,  as  engineer 
in  charge,  perfectly  willing  to  give  you  all  the  technical  assistance  in  my  power 
that  you  might  need  or  ask  for,  I  cannot  undertake  to  give  expert  evidence 
in  this  case  in  favor  of  my  concern,  as  such  evidence  would  be  against  my 
better  convictions  in  this  case.  As,  during  my  last  call  at  your  office,  you 
intimated  my  being  one  of  the  experts,  I  think  it  best  to  let  you  know  as 
early  as  possible  that  I  cannot  undertake  this  duty. 

261 


262  INDUCTION   MOTOR 

Model  maker,  Mr.  W.  J.  Schultz,  called  at  our  office  yesterday  and  I  gave 
him  all  the  necessary  instructions  for  making  the  devices  that  we  had  deemed 
advisable  to  make  for  this  suit.     Mr.  Schultz  is  thus  prepared  to  let  us  have 
his  bid  on  them  and  this  will  be  submitted  to  our  management. 
I  remain, 

Yours  very  truly, 
(Signed)  B.  A.  BEHREND, 
CHIEF  ENGINEER,  ETC. 

It  was  a  matter  of  gratification  to  the  writer  that  almost  20 
years  later,  when  he  was  a  member  of  the  Edison  Medal  Com- 
mittee, he  was  able  to  propose  the  name  of  Tesla  for  the  award 
of  the  Edison  Medal  and  upon  the  occasion  of  the  presentation 
of  the  Medal  to  express  his  great  admiration  for  the  medallist's 
creative  work.  There  is  published  herewith,  as  the  closing 
chapter  of  a  long  story,  the  writer's  address  as  it  was  delivered 
on  that  occasion. 

To  PROFESSOR  ANDRE  BLONDEL 

OF  L'ECOLE  NATIONALS  DES  FONTS  ET  CHAUSSEES 

OF  PARIS 

An  honorary  member  of  our  Institute,  whose  brilliant  and  inspiring  work 
a  generation  ago  laid  the  theoretical  foundation  of  the  development  of  alternating- 
current  engineering  practice;  to  whose  generous  support  of  my  own  modest 
labors,  over  a  score  of  years  ago,  I  owed  recognition;  in  the  hour  of  his  country's 
trial,  I  beg  leave  to  inscribe  these  remarks.  Hands  across  the  sea,  may  a 
happier  future  dawn  before  us! 

B.  A.  BEHREND. 

Address 

Mr.  Chairman:  Mr.  President  of  the  American  Institute  of  Electrical  Engi- 
neers: Fellow  Members:  Ladies  and  Gentlemen: 

DY  an  extraordinary  coincidence,  it  is  exactly  twenty-nine  years  ago,  to 
the  very  day  and  hour,  that  there  stood  before  this  Institute  Mr.  Nikola 
Tesla,  and  he  read  the  following  sentences: 

"To  obtain  a  rotary  effort  in  these  motors  was  the  subject  of  long  thought. 
In  order  to  secure  this  result  it  was  necessary  to  make  such  a  disposition 
that  while  the  poles  of  one  element  of  the  motor  are  shifted  by  the  alternate 
currents  of  the  source,  the  poles  produced  upon  the  other  elements  should 
always  be  maintained  in  the  proper  relation  to  the  former,  irrespective  of 
the  speed  of  the  motor.  Such  a  condition  exists  in  a  continuous  current 
motor:  but  in  a  synchronous  motor,  such  as  described,  this  condition  is 
fulfilled  only  when  the  speed  is  normal. 

"The  object  has  been  attained  by  placing  within  the  ring  a  properly 
subdivided  cylindrical  iron  core  wound  with  several  independent  coils 


APPENDIX  263 

closed  upon  themselves.  Two  coils  at  right  angles  are  sufficient,  but  a 
greater  number  may  be  advantageously  employed.  It  results  from  this 
disposition  that  when  the  poles  of  the  ring  are  shifted,  currents  are  generated 
in  the  closed  armature  coils.  These  currents  are  the  most  intense  at  or  near 
the  points  of  the  greatest  density  of  the  lines  of  force,  and  their  effect  is  to 
produce  poles  upon  the  armature  at  right  angles  to  those  of  the  ring,  at 
least  theoretically  so;  and  since  this  action  is  entirely  independent  of  the 
speed — that  is,  as  far  as  the  location  of  the  poles  is  concerned — a  continuous 
pull  is  exerted  upon  the  periphery  of  the  armature.  In  many  respects 
these  motors  are  similar  to  the  continuous  current  motors.  If  load  is  put  on, 
the  speed,  and  also  the  resistance  of  the  motor,  is  diminished  and  more 
current  is  made  to  pass  through  the  energizing  coils,  thus  increasing  the 
effort.  Upon  the  load  being  taken  off,  the  counter-electromotive  force 
increases  and  less  current  passes  through  the  primary  or  energizing  coils. 
Without  any  load  the  speed  is  very  nearly  equal  to  that  of  the  shifting  poles 
of  the  field  magnet. 

"It  will  be  found  that  the  rotary  effort  in  these  motors  fully  equals  that  of 
the  continuous  current  motors.  The  effort  seems  to  be  greatest  when  both 
armature  and  field  magnet  are  without  any  projections." 

Not  since  the  appearance  of  Faraday's  "Experimental  Researches  in 
Electricity"  has  a  great  experimental  truth  been  voiced  so  simply  and  so 
clearly  as  this  description  of  Mr.  Tesla's  great  discovery  of  the  generation 
and  utilization  of  poly-phase  alternating  currents.  He  left  nothing  to  be 
done  by  those  who  followed  him.  His  paper  contained  the  skeleton  even  of 
the  mathematical  theory. 

Three  years  later,  in  1891,  there  was  given  the  first  great  demonstration, 
by  Swiss  engineers,  of  the  transmission  of  power  at  30,000  volts  from  Lauffen 
to  Frankfort  by  means  of  Mr.  Tesla's  system.  A  few  years  later  this  was 
followed  by  the  development  of  the  Cataract  Construction  Company,  under 
the  presidency  of  our  member,  Mr.  Edward  D.  Adams,  and  with  the  aid  of 
the  engineers  of  the  Westinghouse  Company.  It  is  interesting  to  recall  here 
to-night  that  in  Lord  Kelvin's  report  to  Mr.  Adams,  Lord  Kelvin  recom- 
mended the  use  of  direct  current  for  the  development  of  power  at  Niagara 
Falls  and  for  its  transmission  to  Buffalo. 

The  due  appreciation  or  even  enumeration  of  the  results  of  Mr.  Tesla's 
inventions  is  neither  practicable  nor  desirable  at  this  moment.  There  is  a 
time  for  all  things.  Suffice  it  to  say  that,  were  we  to  seize  and  to  eliminate 
from  our  industrial  world  the  results  of  Mr.  Tesla's  work,  the  wheels  of 
industry  would  cease  to  turn,  our  electric  cars  and  trains  would  stop,  our 
towns  would  be  dark,  our  mills  would  be  dead  and  idle.  Yea,  so  far  reaching 
is  this  work,  that  it  has  become  the  warp  and  woof  of  industry. 

The  basis  for  the  theory  of  the  operating  characteristics  of  Mr.  Tesla's 
rotating  field  induction  motor,  so  necessary  to  its  practical  development,  was 
laid  by  the  brilliant  French  savant,  Professor  Andre"  Blondel,  and  by  Profes- 
sor Kapp  of  Birmingham.  It  fell  to  my  lot  to  complete  their  work  and  to 
coordinate — by  means  of  the  simple  "circle  diagram" — the  somewhat 
mysterious  and  complex  experimental  phenomena.  As  this  was  done 
twenty-one  years  ago,  it  is  particularly  pleasing  to  me,  upon  the  coming 
of  age  of  this  now  universally  accepted  theory — tried  out  by  application  to 


264  INDUCTION  MOTOR 

several  million  horsepower  of  machines  operating  in  our  great  industries — 
to  pay  my  tribute  to  the  inventor  of  the  motor  and  the  system  which  have 
made  possible  the  electric  transmission  of  energy.  His  name  marks  an 
epoch  in  the  advance  of  electrical  science.  From  that  work  has  sprung  a 
revolution  in  the  electrical  art. 

We  asked  Mr.  Tesla  to  accept  this  medal.  We  did  not  do  this  for  the 
mere  sake  of  conferring  a  distinction,  or  of  perpetuating  a  name;  for  so  long 
as  men  occupy  themselves  with  our  industry,  his  work  will  be  incorporated  in 
the  common  thought  of  our  art,  and  the  name  of  Tesla  runs  no  more  risk  of 
oblivion  than  does  that  of  Faraday,  or  that  of  Edison. 

Nor  indeed  does  this  Institute  give  this  medal  as  evidence  that  Mr.  Tesla's 
work  has  received  its  official  sanction.  His  work  stands  in  no  need  of  such 
sanction. 

No,  Mr.  Tesla,  we  beg  you  to  cherish  this  medal  as  a  symbol  of  our 
gratitude  for  a  new  creative  thought,  the  powerful  impetus,  akin  to  revolu- 
tion, which  you  have  given  to  our  art  and  to  our  science.  You  have  lived 
to  see  the  work  of  your  genius  established.  What  shall  a  man  desire  more 
than  this?  There  rings  out  to  us  a  paraphrase  of  Pope's  lines  on  Newton: 

Nature  and  Nature's  laws  lay  hid  in  night: 
God  said,  "Let  TESLA  be,"  and  all  was  light. 

New  York  City 

May  18,  1917. 


INDEX 

Reference  to  Pages 


Adams,  Comfort,  A., 114,  146 

Adams,  Edward  D., 263 

Alexanderson,  E.  F.  W., 179,  181,  204,  250 

and  Hill 258 

on  U.  S.  Battleship  New  Mexico 115 

Altes,  W.  C.  K., 145,  152 

Arnold,  E.,  and  J.  L.  la  Cour 11,  19,  114,  134,  145,  152,  175 

Ampere-turns 57 

Atkinson,  L.  B . , 250 

B 

Bedell,  F.,  early  papers 5,  8,  9 

Behn-Eschenburg,  H., 18,  111,  112,  113,  254 

his  connection  of  shunt  motor  fields 255 

Behrend,  B.  A 203,  204 

criticism  of  his  leakage  factor  by  Hobart 110 

earliest  paper  on  discovery  and  proof  of  circle  diagram 6 

excentric  magnetic  pull , 198 

on  single-phase  system  in  Gassier 's  Magazine 248 

stray-coefficients 24 

"the  Debt  of  Electrical  Engineering  to  C.  E.  L.  Brown," 3 

BSthenod,  J 11 

Blakesley,  Thomas  H.,  Reference  to  early  papers 4,  5 

Blondel,  Andre 146,  149,  152,  233 

Blondel's  stray-coefficients 24 

early  papers 6 

references , 

takes  up  Behrend's  circle  diagram 7 

Bouasse,  Henri, 197 

Boy  de  la  Tour,  Henri 10 

Bragstad,  O.  S 152 

Bridges,  effect  of  slot  bridges  on  short  circuit  current 88 

Brown,  Boveri  Company 178,  187 

Brown,  C.  E.  L 1 

"Debt  of  Electrical  Engineering  to," 3 

squirrel  cage  induction  motor  of  1891 3 

Burson,  H.  A., 261 

265 


266  INDEX 


Cassier's  Magazine,  quotation  from 248 

V  Circle  diagram,  first  experimental  corroboration  of 94 

for  air-gaps  of  different  lengths 93 

for  different  frequencies, ; . .  IQg 

for  series  polyphase  commutator  motor •.  143 

Commutation,  methods  of  improving 253 

Commutator  motors,  polyphase,  properties  of 124 

comparison  with  squirrel  cage 128 

local  leakage  reactance  of 126 

shunt  polyphase 146 

types  of  variable  speed .  182 

variable  and  constant  secondary  reactance  of 131 

Compensated  repulsion  motor 240 

diagram  of  connections 241 

time  phase  diagram 241 

Concatenation 153 

comparison  of  group  with  single  motor 173 

equivalent  circuits 155,  169 

polar  loci 162,  163,  166,  168 

torque  curves 161 

vector  diagrams 157 

Constant  current  transformer,  historical  reference 5 

circle  diagram  of . '. 25 

Constant  voltage  transformer 27 

Copper  loss,  accounting  for  secondary 42 

accounting  for  primary 43 

Correction  of  power  factor 188 

Greedy,  F., 255 

his  motor 250 

Crehore,  A.  C.,  see  also  Bedell,  F 5 

Cross-flux  theory,  of  single-phase  induction  motor 216 

equivalent  circuits  of 228 

D 

Danielson,  E.,  early  motor  design 7 

-Burke 197 

Darwin,  Sir  George  Howard Preface 

Dead  points  in  torque  curve 79 

Deri-Latour  motor 249 

Dobrowolsky,  M.  von  Dolivo-, 1,  115 

term  "wattless  current," 20 

term  "wattless  component," 29 

Double  squirrel  cage  induction  motor 115 

flux  diagram  of 1 117 

polar  diagram  of 120 

torque  curves  of 121,  122 


INDEX  267 

Doubly-fed  motors 147,  249 

Dreyfus,  L.,  and  F.  Hillebrand 134,  152 

Dudley,  A.  M., Preface 

E 

Edison  medal,  address  of  B.  A.  Behrend 262 

Eichberg,  F 152,  175 

compensated  repulsion  motor  of 240 

Eickemeyer,  R., 246 

Elementary  theory  of  induction  motor 64  et  seq. 

Equivalent  circuits,  comparison  of  squirrel  cage  with  commutator  rotor  133 

for  double  squirrel  cage  induction  motor 117 

in  concatenation 155,  169 

of  single-phase  induction  motor 208,  209 

of  single-phase  induction  motor  in  the  cross-flux  theory 228 

F 

Fechheimer,  Carl  J. , 261 

Ferranti,  Sebastian  Ziani  de,  10,000  volt  line  Deptford  to  London 5 

Ferraris,  Galileo 204 

Field,  A.  B.,  Preface 258,  261 

Field  belt 57 

Fortescue,  C.  Le  G. , 258,  260 

Frankfort-on-the-Main,  Electrical  Exposition,  1891 1 

Frequency,  drawbacks  of  high 107 

Fresnel's  theorem 204 

Fynn,  Val. , 249 

G 

Generator,  see  Induction  Generator 82 

Gilman,  R.  E 260 

Goerges,  H 137,  152 

his  shunt  poly-phase  commutator  motor 182 

Goldschmidt,  R 114 

Gray,  Alexander  Miller,  Preface 114,  203,  261 

Use  of  Behrend's  stray-coefficients 23 

Guillet,  A 197 

H 

Hall,  David 261 

Harmonics,  in  flux  belt 78 

Heaviside,  Oliver 1 

"Algebraization," 4 

Hellmund,  R.  E.,  Preface 114 

Translation  of  Heyland's  paper 9,  12 

Helmholtz,  quoted 110 

Hertz,  Heinrich 1 


268  INDEX 

Heyland,  Alexander 7,  13,  152,  177 

his  compensated  induction  motor 182 

Hobart,  H.  M.,  on  leakage  factor 110,  111,  112 

Hodograph  of  induced  voltages  in  distributed  windings 61 

Hopkinson,  Dr.  John 23 

Huxley,  T.  H Preface 

I 

i  Ideas,  forced  and  natural 248 

\  I  Induction  motor,  elementary  theory  of 63 

\J  I   equivalent  to  transformer 73 

\J    torque  of 74 

v/   with  commutator  rotor,  circle  diagram  of 132 

Induction  generator 82  et  seq. 

tests  on 85 

Interpole  shunts 254 

Interpoles  in  single-phase  motors 254 

J 

Jackson,  D.  C Preface 

Johnson,  John  Butler Preface 

K 

Kapp,  Gisbert 196 

definition  of  vopen  circuit  current 29 

"Electrician"  paper,  1890 5 

"Electric  Transmission  of  Energy,"  4th  edition 6 

his  vibrator 191 

Institution  of  Civil  Engineers  paper,  1885 4 

on  elementary  theory  of  induction  motor 63 

Karapetoff,  V 151 

the  secomor 145 

Kelvin's  report  on  Cataract  Construction  Co.'s  Development 263 

Kirchhoffs  laws  applied  to  squirrel  cage 70,  71 

Kittler,  E.,  and  W.  Petersen 12,  152 

Kraemer  system 176 

Krug,  Dr.  Karl 11 

L 

La  Cour,  J.  L 134,  176,  183 

Lamb,  Horace .  •  •  •    196 

Lamme,  B.  G.. 1,  136,  179,  186,  246,  247,  253 

concatenation •   175 

quoted 107 

"The  Story  of  the  Induction  Motor,"  quoted 4 

Latour,  M.  C.  A 250 

compensated  repulsion  motor  of 240 

his  motor,  tests  of 251 


INDEX  269 

Lawrence,  R.  R 26 

Leakage  coefficient 26 

Leakage  factor 86,  104 

as  affected  by  frequency 107 

for  different  numbers  of  poles 106 

historical  and  critical  discussion  of 110 

influence  of  pole-pitch  on 99 

of  single-phase  induction  motor 210 

the  effect  of  the  air-gap  on 95 

Leakage  flux,  conventional  but  erroneous  representation 102 

Leakage  paths  of  double  squirrel  cage  motor 116 

Leblanc,  Maurice 187,  189 

Scherbius  rotor- 189 

Lehmann 11 

Load  loss 94 

Locked  current,  see  short  circuit  current 86 

Loss,  iron,  as  shown  in  equivalent  circuits 44 

Loss  lines 42,  43 

V  Losses,  as  represented  in  the  circle  diagram 45 

Lydall,  F 176 

M 

Magnetic  pull,  unbalanced 198 

Magnetizing  current  as  affected  by  high  tooth  induction 60 

Mailloux,  C.  0 10 

Maxwell,  James  Clerk. 1 

Me 

McAllister,  A.  S 114,  217,  216,  252 

his  transformations 49  et  seq. 

McCormick,  Bradley  T 261 

N 

New  Mexico,  United  States  Battleship 78,  115 

Newton,  cited 264 

New  York,  New  Haven,  &  Hartford  railroad 248 

O 

Oerlikon  Company '. 1,  180 

Osnos,  M 183,  233,  239 

Time  phase  diagram  of  compensated  repulsion  motor 243 

Ossanna 11 

P 

Pennsylvania  Railroad 258 

Perry,  John 196 


, 


270  INDEX 

Pertsch,  J.  G .   203 

Pole-pitch,  its  influence  on  the  leakage  factor 99 

Poly-phase  commutators 129,  130 

Poly-phase  commutator  for  generation  of  leading  currents 187 

Potier,  Alfred 18,  216 

Power,  single-phase  from  three-phase  systems 258 

Power  factor,  maximum  power  factor  depending  on  leakage 30 

Prescott,  J 196 

Pupin,  M.  1 82,  211 

Q 

Quarter-phase  system,  e.m.fs.  induced  in 62 

R 

Recuperator 189,  190 

Repulsion  motor,  single-phase 233 

commutation 240 

effect  of  brush-shift 238 

effect  of  resistance 238 

space-phase  diagram . . , 234 

speed  in  the  diagram 237 

time-phase  diagram 236 

torque 237 

Resistance,  primary,  its  effect  on  the  operating  characteristics  of  the 

motor 31 

correction  for  it  by  Behrend's  method 32 

correction  for  it  by  method  of  reciprocal  vectors 38 

correction  for  it  by  Sumec  et  al , 33 

Resistance  leads 253 

Rosenberg,  E 203 

Roth,  Edouard 11,  152 

Routh,  Edward  John 196 

Ruedenberg,  R 177 

Russell,  Alexander Preface 

S 

Saturation,  affects  short-circuit  current 87 

necessity  of,  in  series  poly-phase  commutator  motor 145 

Scherbius,  A 145,  176 

Schrage,  D.  H.  K 184 

Scott,  Charles  F 1 

Series  poly-phase  commutator  motor 137 

brush-shift  in 140 

slip  of 142 

space-phases  and  time-phases 139 

torque  of 141 

Shallenberger 1 


«Aai 


INDEX  271 


i/ 


Short-circuit  current 86 

affected  by  saturation 87 

circle  diagram  of 148 

error  introduced  by  its  use 94 

Shunt  poly-phase  commutator  motor  total  current  of 150 

equivalent  model  of 151 

Shuttleworth,  N 152 

Single-phase  commutator  motors 246 

different  types  of 249 

Single-phase  induction  motor 204 

circle  diagram  of 207 

cross  flux  theory  of 216 

equivalent  circuits  of 208 

experimental  data  of 213 

leakage  factor  of 210 

magnetizing  current 214 

magnetizing  current  of 205 

no  load  current  of 210 

two-motor  theory 205 

Single-phase  secondary,  poly-phase  motor  with 173 

Slepian,  J 216 

Slip,  determination  of 75 

in  the  elementary  theory 66 

in  single-phase  induction  motor 211 

Slip-ring  type,  comparison  with  commutator  type 135 

Slots,  closed 91 

closed,  short-circuit  current  with 92 

in  double  squirrel  cage  motor 116 

number  of  slots 89 

open  and  closed,  affect  short-circuit  current 87 

.Space-phase,  interchangeable  with  time-phase 74 

\s  Speed  control,  methods  of 153 

Squirrel  cage,  effect  of  end  rings 73 

first  squirrel  cage  motor  of  C.  E.  L.  Brown's 3 

in  the  elementary  theory 66 

theory  of 69 

Stability  of  induction  machine  as  motor  or  generator 83 

Steinmetz,  C.  P 246,  247 

Sumec,  J.  K 18,  201,  203 

Formula  of  excentric  magnetic  pull  by 201 

his  circles  of  single-phase  induction  motor 224,  225,  226 

Uses  Behrend's  stray-coefficient 24 

T 

Tesla,  Nikola 1,  107 

his  patents 261 

Tests  of  20  H.P.  motor  made  in  1896 93 

Thomaelen 10,  18,  19 


/ 


272  INDEX 

Thompson,  Silvanus  P 14,  16 

on  leakage  factor m 

Time-phase  interchangeable  with  space-phase 74 

Torque  curves  of  series  poly-phase  commutator  motor 144 

Torque,  dead  points  in 79 

in  single-phase  induction  motor 211 

of  induction  motor 74 

Transformer,  equivalent  to  induction  motor 73 

with  capacity  load 47 

with  inductive  load 46 

V 

Vectors,  reciproca. 38  et  seq. 

term  misued  for  e.m.f.  and  current 4 

Vibrator 191 

theory  of 192 

W 

Wagner-Fynn  motor ' 249 

Walker,  Miles 16,  176,  258 

Washington,  Baltimore  &  Annapolis  single-phase  railway 246 

Waters,  W.  L 261 

Weaver,  William  D Preface 

Westinghouse  Company,  Kapp  vibrator  built  by 194 

their  single-phase  motor 253 

Westinghouse,  George 247 

Wightman,  M.  J 247 

compensated  repulsion  motor  of 240 

Williamson,  R.  B 261 

Wilson,  British  patent  of 137 


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UNIVERSITY  OF  CALIFORNIA  LIBRARY 


